The Concept of Congruence Module two Grade 8 The Concept of Congruence Module two

Congruence and Angle Relationships TOPIC C Congruence and Angle Relationships

Definition of congruence and some basic properties 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Definition of congruence and some basic properties Lesson eleven

Lesson eleven 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Example one A geometric figure 𝑆 is said to be congruent to another geometric figure 𝑆′ if there is a sequence of rigid motions that maps 𝑆 to ′ , i.e., Congruence(𝑆) = 𝑆′ . The notation related to congruence is the symbol ≅. When two figures are congruent, like 𝑆 and 𝑆′, we can write: 𝑆𝑆 ≅ 𝑆𝑆′. We want to describe the sequence of rigid motions that demonstrates the two triangles shown below are congruent, i.e., △ 𝐴BC ≅ △ 𝐴′ 𝐵′ 𝐶′ .

Lesson eleven 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Example one

Lesson eleven 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Example two

Lesson eleven 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Notes A basic rigid motion maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle. A basic rigid motion preserves lengths of segments. A basic rigid motion preserves measures of angles.

Lesson eleven 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE ONE Describe the sequence of basic rigid motions that shows 𝑆1 ≅ 𝑆2. Describe the sequence of basic rigid motions that shows 𝑆2 ≅ 𝑆3. Describe a sequence of basic rigid motions that shows 𝑆1 ≅ 𝑆3.

Properties of Congruence Lesson eleven 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Properties of Congruence (Congruence 1) A congruence maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle. (Congruence 2) A congruence preserves lengths of segments. (Congruence 3) A congruence preserves measures of angles.

Lesson eleven 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE TWO Perform the sequence of a translation followed by a rotation of Figure XYZ, where 𝑇 is a translation along a vector AB�⃗, and 𝑅 is a rotation of 𝑑 degrees (you choose 𝑑) around a center 𝑂. Label the transformed figure 𝑋′ 𝑌′ 𝑍′ . Will XYZ≅ 𝑋′ 𝑌′ 𝑍′ ?

Lesson Summary Lesson eleven NYS COMMON CORE MATHEMATICS CURRICULUM 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson Summary

Angles associated with parallel lines 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Angles associated with parallel lines Lesson twelve

EXPLORATORY CHALLENGE 1 Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXPLORATORY CHALLENGE 1 In the figure below, 𝐿1 is not parallel to 𝐿2, and 𝑚 is a transversal. Use a protractor to measure angles 1–8. Which, if any, are equal? Explain why. (Use your transparency if needed.)

Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Discussion Questions What did you notice about the pairs of angles in the first diagram when the lines, 𝐿1 and 𝐿2, were not parallel? Why are vertical angles equal in measure? Angles that are on the same side of the transversal in corresponding positions (above each of 𝐿1 and 𝐿2 or below each of 𝐿1 and 𝐿2) are called corresponding angles. Name a pair of corresponding angles in the diagram. When angles are on opposite sides of the transversal and between (inside) the lines 𝐿1 and 𝐿2, they are called alternate interior angles. Name a pair of alternate interior angles. When angles are on opposite sides of the transversal and outside of the parallel lines (above 𝐿1 and below 𝐿2), they are called alternate exterior angles. Name a pair of alternate exterior angles.

EXPLORATORY CHALLENGE 2 Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXPLORATORY CHALLENGE 2 In the figure below, 𝐿1 ∥ 𝐿2, and 𝑚 is a transversal. Use a protractor to measure angles 1–8. List the angles that are equal in measure.

EXPLORATORY CHALLENGE 2 Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXPLORATORY CHALLENGE 2 What did you notice about the measures of ∠1 and ∠5? Why do you think this is so? (Use your transparency if needed.) What did you notice about the measures of ∠3 and ∠7? Why do you think this is so? (Use your transparency if needed.) Are there any other pairs of angles with this same relationship? If so, list them. c. What did you notice about the measures of ∠4 and ∠6? Why do you think this is so? (Use your transparency if needed.) Is there another pair of angles with this same relationship?

Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Discussion Questions Were the vertical angles in Exploratory Challenge 2 equal like they were in Exploratory Challenge 1? Why? What other angles were equal in the second diagram when the lines 𝐿1 and 𝐿2 were parallel? Let’s look at just ∠1 and ∠5. What kind of angles are these, and how do you know? We have already said that these two angles are equal in measure. Who can explain why this is so?

Discussion Questions What did you notice about ∠3 and ∠7? Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Discussion Questions What did you notice about ∠3 and ∠7? What other pairs of corresponding angles are in the diagram? In Exploratory Challenge 1, the pairs of corresponding angles we named were not equal in measure. Given the information provided about each diagram, can you think of why this is so?

Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Discussion Questions Are ∠4 and ∠6 corresponding angles? If not, why not? What kind of angles are ∠4 and ∠6? How do you know? We have already said that ∠4 and ∠6 are equal in measure. Why do you think this is so?

Discussion Questions Name another pair of alternate interior angles. Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Discussion Questions Name another pair of alternate interior angles. In Exploratory Challenge 1, the pairs of alternate interior angles we named were not equal in measure. Given the information provided about each diagram, can you think of why this is so? Are ∠1 and ∠7 corresponding angles? If not, why not? Are ∠1 and ∠7 alternate interior angles? If not, why not? What kind of angles are ∠1 and ∠7?

Discussion Questions Name another pair of alternate exterior angles. Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Discussion Questions Name another pair of alternate exterior angles. These pairs of alternate exterior angles were not equal in measure in Exploratory Challenge 1. Given the information provided about each diagram, can you think of why this is so?

Theorem and its Converse Lesson twelve 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Theorem and its Converse Theorem: When parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent, the pairs of alternate interior angles are congruent, and the pairs of alternate exterior angles are congruent The converse of the theorem states that if you know that corresponding angles are congruent, then you can be sure that the lines cut by a transversal are parallel.

Lesson Summary Lesson twelve NYS COMMON CORE MATHEMATICS CURRICULUM 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson Summary

Angle sum of a triangle Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Angle sum of a triangle Lesson thirteen

Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Notes The angle sum theorem for triangles states that the sum of the interior angles of a triangle is always 180° (∠ sum of △). It does not matter what kind of triangle it is (i.e., acute, obtuse, right); when you add the measure of the three angles, you always get a sum of 180°.

Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Notes

Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Notes We want to prove that the angle sum of any triangle is 180°. To do so, we will use some facts that we already know about geometry: A straight angle is 180° in measure. Corresponding angles of parallel lines are equal in measure (corr. ∠𝑠, � 𝐴B ∥ 𝐶D). Alternate interior angles of parallel lines are equal in measure (alt. ∠𝑠, 𝐴B ∥ 𝐶D).

EXPLORATORY CHALLENGE 1 Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXPLORATORY CHALLENGE 1 Let triangle ABC be given. On the ray from 𝐵 to 𝐶, take a point 𝐷 so that 𝐶 is between 𝐵 and 𝐷. Through point 𝐶, draw a line parallel to AB, as shown. Extend the parallel lines AB and CE. Line AC is the transversal that intersects the parallel lines.

EXPLORATORY CHALLENGE 1 - questions Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXPLORATORY CHALLENGE 1 - questions Name the three interior angles of triangle ABC. Name the straight angle. What kinds of angles are ∠ABC and ∠ECD? What does that mean about their measures? What kinds of angles are ∠BAC and ∠ECA? What does that mean about their measures? We know that ∠BCD = ∠BCA + ∠ECA + ∠ECD = 180°. Use substitution to show that the three interior angles of the triangle have a sum of 180°

EXPLORATORY CHALLENGE 2 Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXPLORATORY CHALLENGE 2 The figure below shows parallel lines 𝐿1 and 𝐿2. Let 𝑚 and 𝑛 be transversals that intersect 𝐿1 at points 𝐵 and 𝐶, respectively, and 𝐿2 at point 𝐹, as shown. Let 𝐴 be a point on 𝐿1 to the left of 𝐵, 𝐷 be a point on 𝐿1 to the right of 𝐶, 𝐺 be a point on 𝐿2 to the left of 𝐹, and 𝐸 be a point on 𝐿2 to the right of 𝐹.

EXPLORATORY CHALLENGE 2 - questions Lesson thirteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXPLORATORY CHALLENGE 2 - questions Name the triangle in the figure. Name a straight angle that will be useful in proving that the sum of the interior angles of the triangle is 180°. Write your proof below.

Lesson Summary Lesson thirteen NYS COMMON CORE MATHEMATICS CURRICULUM 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson Summary

More on angles of a triangle 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM More on angles of a triangle Lesson fourteen

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM DISCUSSION

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM DISCUSSION

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISES 1-4 Name an exterior angle and the related remote interior angles. Name a second exterior angle and the related remote interior angles. Name a third exterior angle and the related remote interior angles. Show that the measure of an exterior angle is equal to the sum of the related remote interior angles.

Find the measure of angle 𝑥. Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE ONE Find the measure of angle 𝑥.

Find the measure of angle 𝑥. Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE TWO Find the measure of angle 𝑥.

Find the measure of angle 𝑥. Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE THREE Find the measure of angle 𝑥.

Find the measure of angle 𝑥. Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXAMPLE FOUR Find the measure of angle 𝑥.

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE FIVE Find the measure of angle 𝒙. Present an informal argument showing that your answer is correct.

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE SIX Find the measure of angle 𝒙. Present an informal argument showing that your answer is correct.

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE SEVEN Find the measure of angle 𝒙. Present an informal argument showing that your answer is correct.

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE EIGHT Find the measure of angle 𝒙. Present an informal argument showing that your answer is correct.

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE NINE Find the measure of angle 𝒙. Present an informal argument showing that your answer is correct.

Lesson fourteen 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM EXERCISE TEN Find the measure of angle 𝒙. Present an informal argument showing that your answer is correct.

Lesson Summary Lesson fourteen NYS COMMON CORE MATHEMATICS CURRICULUM 8.2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson Summary