Illustrative Math Alignment: Grade 8 Unit 1
Rigid Transformations and Congruence
Lesson 16: Parallel Lines and the Angles in a Triangle
Use the following Media4Math resources with this Illustrative Math lesson.
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Math Example--Right Triangles-- Example 1 | Math Example--Right Triangles-- Example 1TopicRight Triangles DescriptionThis example presents a right triangle with sides of length 8 and 10, and an unknown hypotenuse labeled c. The task is to find the value of c using the Pythagorean Theorem. By applying the formula a2 + b2 = c2, we can calculate that c = √(102 + 82) = √(164) = 2 * √(41). |
Right Triangles |
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Math Example--Right Triangles-- Example 10 | Math Example--Right Triangles-- Example 10TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 11 | Math Example--Right Triangles-- Example 11TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 12 | Math Example--Right Triangles-- Example 12TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 13 | Math Example--Right Triangles-- Example 13TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 14 | Math Example--Right Triangles-- Example 14TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 15 | Math Example--Right Triangles-- Example 15TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 16 | Math Example--Right Triangles-- Example 16TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 17 | Math Example--Right Triangles-- Example 17TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 18 | Math Example--Right Triangles-- Example 18TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 19 | Math Example--Right Triangles-- Example 19TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 2 | Math Example--Right Triangles-- Example 2TopicRight Triangles DescriptionIn this example, we have a right triangle with sides of length 3 and 4, and an unknown hypotenuse labeled c. The goal is to determine the value of c using the Pythagorean Theorem. By applying the formula a2 + b2 = c2, we can calculate that c = √(32 + 42) = √(25) = 5. This example builds upon the previous one, reinforcing the application of the Pythagorean Theorem in right triangles. It demonstrates how the theorem can be used with different side lengths, helping students understand its versatility in solving various right triangle problems. |
Right Triangles |
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Math Example--Right Triangles-- Example 20 | Math Example--Right Triangles-- Example 20TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 21 | Math Example--Right Triangles-- Example 21TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 22 | Math Example--Right Triangles-- Example 22TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 23 | Math Example--Right Triangles-- Example 23TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 24 | Math Example--Right Triangles-- Example 24TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 25 | Math Example--Right Triangles-- Example 25TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 26 | Math Example--Right Triangles-- Example 26TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 3 | Math Example--Right Triangles-- Example 3TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 4 | Math Example--Right Triangles-- Example 4TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 5 | Math Example--Right Triangles-- Example 5TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 6 | Math Example--Right Triangles-- Example 6TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 7 | Math Example--Right Triangles-- Example 7TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 8 | Math Example--Right Triangles-- Example 8TopicRight Triangles |
Right Triangles |
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Math Example--Right Triangles-- Example 9 | Math Example--Right Triangles-- Example 9TopicRight Triangles |
Right Triangles |
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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 |
