Illustrative Math Alignment: Grade 8 Unit 1
Rigid Transformations and Congruence
Lesson 15: Adding the Angles in a Triangle
Use the following Media4Math resources with this Illustrative Math lesson.
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Math Example--Polygons--Triangle Classification: Example 24 | Math Example--Polygons--Triangle Classification: Example 24TopicTriangles DescriptionThis example shows a triangle with angles labeled as x, 0.25x, and 1.1x. Solving for x, we find that all angles are less than 90°, classifying the triangle as acute. Triangles are classified by their angle measures or side lengths, with acute triangles having all angles less than 90°. Reviewing examples like this helps students reinforce the criteria for classifying triangles based on angle sizes. Seeing multiple worked-out examples builds students' confidence in applying classification rules and identifying triangle types based on consistent properties. |
Definition of a Triangle |
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Math Example--Polygons--Triangle Classification: Example 3 | Math Example--Polygons--Triangle Classification: Example 3TopicTriangles DescriptionThis example shows a triangle with side lengths of 12, 18, and x, where x is not equal to 12 or 18. The triangle is classified as scalene because all three side lengths are different. Triangles are categorized by side length or angle. Scalene triangles, with all sides unequal, demonstrate a key classification type. Multiple examples help students recognize classification patterns for triangles based on side length or angle measure. Observing various worked-out examples reinforces consistent application of classification criteria, building student confidence in identifying triangle types. |
Definition of a Triangle |
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Math Example--Polygons--Triangle Classification: Example 4 | Math Example--Polygons--Triangle Classification: Example 4TopicTriangles DescriptionThis example displays a triangle with angles measuring 89°, 44°, and 47°. Since all angles are less than 90°, the triangle is classified as acute. Triangle classification can be based on side length or angle measures. Acute triangles have all angles less than 90°, as shown in this example. Reviewing multiple examples in this way helps students visually distinguish triangle types by angle. Providing varied worked-out examples solidifies students’ understanding of the criteria for each triangle type, enabling them to accurately identify triangles by their angle measures. |
Definition of a Triangle |
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Math Example--Polygons--Triangle Classification: Example 5 | Math Example--Polygons--Triangle Classification: Example 5TopicTriangles DescriptionThis example presents a triangle with angles of 48°, 50°, and an unknown angle x. Solving for x, we find that x = 82°, meaning all angles are less than 90°, classifying it as acute. Triangles can be identified by their angles or side lengths. Acute triangles have all angles less than 90°, as seen here. Reviewing examples like this supports student understanding of angle-based classification. Seeing multiple worked examples helps students recognize patterns and confidently classify triangles based on established criteria. |
Definition of a Triangle |
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Math Example--Polygons--Triangle Classification: Example 6 | Math Example--Polygons--Triangle Classification: Example 6TopicTriangles DescriptionThis example shows a triangle with angles of 36°, 100°, and 44°. Because one of the angles is greater than 90°, this triangle is classified as obtuse. Triangles are classified by side lengths or angles. An obtuse triangle has one angle greater than 90°, as seen here. This collection of examples supports students in identifying and classifying triangles based on distinct angle measures. Seeing multiple worked-out examples reinforces students' understanding of classification rules, making it easier to recognize triangle types based on angle or side length. |
Definition of a Triangle |
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Math Example--Polygons--Triangle Classification: Example 7 | Math Example--Polygons--Triangle Classification: Example 7TopicTriangles DescriptionThis example presents a triangle with two known angles, 20° and 50°, and an unknown angle x. Solving for x, we find it to be 110°. Since one angle is greater than 90°, the triangle is classified as obtuse. Triangles can be classified by their angles, where obtuse triangles have one angle over 90°. Reviewing examples like this aids students in understanding classification based on angle measures. Seeing multiple examples reinforces the rules and helps students apply classification consistently and accurately. |
Definition of a Triangle |
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Math Example--Polygons--Triangle Classification: Example 8 | Math Example--Polygons--Triangle Classification: Example 8TopicTriangles DescriptionThis example shows a triangle with two angles, each measuring 60°. Since two angles are congruent, this triangle is classified as isosceles. Triangles can be categorized by side or angle properties. Isosceles triangles have two equal sides or two equal angles, as illustrated here. This collection of examples allows students to observe classifications and understand patterns in triangle properties. Examining multiple examples helps students confidently identify triangle types based on consistent characteristics. |
Definition of a Triangle |
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Math Example--Polygons--Triangle Classification: Example 9 | Math Example--Polygons--Triangle Classification: Example 9TopicTriangles DescriptionThis example shows a triangle with two angles labeled x and one angle labeled y. Since two angles are congruent, the triangle is classified as isosceles. Triangles can be classified by side lengths or angle measures. Isosceles triangles, with two equal sides or angles, are common. Observing various examples helps students understand these classification criteria. Providing multiple examples allows students to identify patterns in side and angle relationships, reinforcing accurate triangle classification. |
Definition of a Triangle |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 1 | Equations Using the Exterior Angle Theorem--Example 1TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 10 | Equations Using the Exterior Angle Theorem--Example 10TopicEquations DescriptionThis example illustrates a more complex application of the Exterior Angle Theorem in solving triangle-related equations. In this scenario, we have a triangle with one known interior angle of 35°, an unknown interior angle y, and an unknown exterior angle x. The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Here, the equation is set up as x = 35° + y. Furthermore, the angles 3x and x are supplementary, allowing you to solve for x. Having solved for x, you can then solve for y. |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 2 | Equations Using the Exterior Angle Theorem--Example 2TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 3 | Equations Using the Exterior Angle Theorem--Example 3TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 4 | Equations Using the Exterior Angle Theorem--Example 4TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 5 | Equations Using the Exterior Angle Theorem--Example 5TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 6 | Equations Using the Exterior Angle Theorem--Example 6TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 7 | Equations Using the Exterior Angle Theorem--Example 7TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 8 | Equations Using the Exterior Angle Theorem--Example 8TopicEquations |
Applications of Triangles |
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Math Example--Solving Equations--Equations Using the Exterior Angle Theorem--Example 9 | Equations Using the Exterior Angle Theorem--Example 9TopicEquations DescriptionThis example presents a more challenging application of the Exterior Angle Theorem in solving triangle-related equations. In this scenario, we have a triangle with one known interior angle of 25°, an unknown interior angle y, and an unknown exterior angle x. The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Here, the equation is set up as x = 25° + y. You can also use the fact that x and 2x are supplementary, allowing you to solve for x. By solving for x, you can then solve for y using the triangle equation. |
Applications of Triangles |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 1 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 1TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 10 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 10TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 2 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 2TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 3 | TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 4 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 4TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 5 | TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 6 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 6TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 7 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 7TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 8 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 8TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Equations with Angles from Parallel Lines Cut by a Transversal--Example 9 | Equations with Angles from Parallel Lines Cut by a Transversal--Example 9TopicEquations |
Parallel Lines, Applications of Angles and Planes and Applications of Equations and Inequalities |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 1 | Solving Equations Using Triangle Properties: Example 1TopicEquations DescriptionThis example focuses on solving equations using the properties of similar isosceles triangles. Isosceles triangles are characterized by having two equal sides and two equal base angles. In this case, we have two similar isosceles triangles, which means they share the same shape but may differ in size. The equation to be solved involves finding the unknown angle x, given that one of the angles is 20°. The property of vertical angles tells us that the angle vertical to the 20° angle is also 20° |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 10 | Solving Equations Using Triangle Properties: Example 10TopicEquations DescriptionThis example, similar to Example 9, involves solving equations using the properties of a kite and applying the exterior angle theorem. We are again given one angle of 40° and two unknown angles, y and x. The goal is to set up and solve equations to find the values of y and x using the properties of kites and the exterior angle theorem. Key properties to consider: |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 2 | Solving Equations Using Triangle Properties: Example 2TopicEquations DescriptionThis example explores solving equations using the properties of similar isosceles triangles, building upon the concepts introduced in Example 1. In this case, we have two similar isosceles triangles with one known angle of 70° and an unknown angle x. The goal is to determine the value of x using triangle properties and algebraic techniques. |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 3 | Solving Equations Using Triangle Properties: Example 3TopicEquations DescriptionThis example focuses on solving equations involving parallel lines cut by a transversal, a fundamental concept in geometry. The problem presents two parallel lines intersected by two transversals that also form a triangle. We are given that one angle measures 120° and the corresponding angle can be expressed as (y + 40)°. The goal is to determine the value of y using the properties of angles formed by parallel lines and a transversal. When parallel lines are cut by a transversal, several important angle relationships are formed: |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 4 | Solving Equations Using Triangle Properties: Example 4TopicEquations DescriptionThis example demonstrates solving equations using the Exterior Angle Theorem in the context of parallel lines cut by a transversal, two crucial concepts in geometry. The problem presents a triangle with two known interior angles of 80° and y, and an unknown exterior angle x°. We are also given that 80 - y = 50, which simplifies to y = 30. The goal is to determine the value of x using the properties of triangles and the Exterior Angle Theorem. The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Wo we get x = 80 + 30, or x = 110. |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 5 | Solving Equations Using Triangle Properties: Example 5TopicEquations DescriptionThis example focuses on solving equations involving isosceles triangles centered in a circle. The problem presents two equations: z - y = 20 and z + y = 120, where z and y represent angles in the isosceles triangles. The goal is to solve this system of equations to find the values of z and y, utilizing properties of isosceles triangles and circles. |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 6 | Solving Equations Using Triangle Properties: Example 6TopicEquations DescriptionThis example explores solving equations involving triangles that share a vertex at the center of a circle. We are presented with two equations: x + y = 75 and z + y = 110, where x, y, and z represent angles in the triangle The goal is to solve this system of equations to find the values of x, y, and z, utilizing properties of isosceles triangles. Since each of the triangles is isosceles, we know that z + 55 + 55 = 180 and therefore, z = 70°. We substitute this into one of the equations: 70 + y = 110 7 = 40° |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 7 | Solving Equations Using Triangle Properties: Example 7TopicEquations DescriptionThis example focuses on solving equations involving isosceles triangles with a common vertex and base. We are given two angles, 62° and 99°, and need to find the unknown angle x. This problem demonstrates how to use the properties of isosceles triangles and the sum of angles in a triangle to solve for an unknown angle. Key properties to consider: |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 8 | Solving Equations Using Triangle Properties: Example 8TopicEquations DescriptionThis example involves solving an equation using the properties of a trapezoid with an embedded parallelogram and applying the exterior angle theorem. We are given two angles, 30° and 110°, and need to find the unknown angle x. This problem demonstrates the application of multiple geometric concepts to solve a complex equation. Key properties to consider: |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations Using Triangle Properties: Example 9 | Solving Equations Using Triangle Properties: Example 9TopicEquations DescriptionThis example focuses on solving equations using the properties of a kite and applying the exterior angle theorem. We are given one angle of 30° and two unknown angles, y and x. The goal is to set up and solve equations to find the values of y and x using the properties of kites and the exterior angle theorem. Key properties to consider: |
Applications of Equations and Inequalities and Applications of Triangles |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 1 | Solving Equations with Angle Measures 2--Example 1TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 10 | Solving Equations with Angle Measures 2--Example 10TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 2 | Solving Equations with Angle Measures 2--Example 2TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 3 | Solving Equations with Angle Measures 2--Example 3TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 4 | Solving Equations with Angle Measures 2--Example 4TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 5 | Solving Equations with Angle Measures 2--Example 5TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 6 | Solving Equations with Angle Measures 2--Example 6TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 7 | Solving Equations with Angle Measures 2--Example 7TopicEquations DescriptionThis example demonstrates solving equations involving angle measures in a kite. A kite has two pairs of adjacent congruent angles. In this case, we have angles represented as (x+50)°, (x+50)°, (y+20)°, and y°. To solve this problem, we apply two key principles: the sum of angles in a quadrilateral is 360°, and the sum of the angles of a triangle is 180°. You can use the triangle equation to solve for y. Once you determine the value for y, you can use that to find x using either the triangle or the quadrilateral equation. In the solution shown, the triangle equation is used. |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 8 | Solving Equations with Angle Measures 2--Example 8TopicEquations DescriptionThis example illustrates solving equations involving angle measures in a kite. The kite has two known angles of 70° and 40°, and two unknown angles represented as (x+y)°. To solve this problem, we look at the triangles formed by one of the diagonals of the kite and use the triangle equation. First solve for x with the top triangle. Once you find x, use that value to solve for y in the bottom triangle. You could also use the quadrilateral equation to solve for y. |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures 2--Example 9 | Solving Equations with Angle Measures 2--Example 9TopicEquations |
Solving Multistep Equations and Applications of Quadrilaterals |
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Math Example--Solving Equations--Solving Equations with Angle Measures--Example 1 | Solving Equations with Angle Measures--Example 1TopicEquations |
Solving Multistep Equations and Applications of Angles and Planes |
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Math Example--Solving Equations--Solving Equations with Angle Measures--Example 10 | Solving Equations with Angle Measures--Example 10TopicEquations |
Solving Multistep Equations and Applications of Angles and Planes |
