Grade 8 The Concept of Congruence Module two. TOPIC C Congruence and Angle Relationships.

Grade 8

The Concept of CongruenceModule two

TOPIC C

Congruence and Angle Relationships

DEFINITION OF CONGRUENCE AND SOME BASIC PROPERTIES

Lesson eleven

NYS COMMON CORE MATHEMATICS CURRICULUM 8.2

NYS COMMON CORE MATHEMATICS CURRICULUM 8.2Lesson eleven

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


Example one


Example two


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.


EXERCISE ONEa. Describe the sequence of basic rigid motions that shows 1 𝑆 ≅

2. 𝑆b. Describe the sequence of basic rigid motions that shows 2 𝑆 ≅

3. 𝑆c. Describe a sequence of basic rigid motions that shows 1 3. 𝑆 ≅ 𝑆


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.


EXERCISE TWOPerform the sequence of a translation followed by a rotation of Figure XYZ, where is a translation along a vector AB , and 𝑇 𝑅�⃗�is a rotation of degrees (you choose ) around a center . 𝑑 𝑑 𝑂Label the transformed figure . Will XYZ ?𝑋′ 𝑌′ 𝑍′ ≅ 𝑋′ 𝑌′ 𝑍′


Lesson Summary

ANGLES ASSOCIATED WITH PARALLEL LINES

Lesson twelve


NYS COMMON CORE MATHEMATICS CURRICULUM 8.2Lesson twelve

EXPLORATORY CHALLENGE 1In the figure below, 𝐿1 is not parallel to 𝐿2, and is a 𝑚transversal. Use a protractor to measure angles 1–8. Which, if any, are equal? Explain why. (Use your transparency if needed.)


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 2In the figure below, 𝐿1 ∥ 𝐿2, and is a transversal. Use a 𝑚protractor to measure angles 1–8. List the angles that are equal in measure.


EXPLORATORY CHALLENGE 2a. What did you notice about the measures of 1 and 5? ∠ ∠

Why do you think this is so? (Use your transparency if needed.)

b. 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?


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?∠ ∠

• 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?


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.

• 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.

• 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• 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

ANGLE SUM OF A TRIANGLE

Lesson thirteen


NYS COMMON CORE MATHEMATICS CURRICULUM 8.2Lesson thirteen

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°.


Notes


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. , B D). ∠ �⃗ ∥𝑠 𝐴 𝐶• Alternate interior angles of parallel lines

are equal in measure (alt. , B D).∠ ∥𝑠 𝐴 𝐶


EXPLORATORY CHALLENGE 1Let triangle ABC be given. On the ray from to , take a 𝐵 𝐶point so that is between and . Through point , draw 𝐷 𝐶 𝐵 𝐷 𝐶a 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 - questionsa. Name the three interior angles of triangle ABC.

b. Name the straight angle.

c. What kinds of angles are ABC and ECD? What does that ∠ ∠mean about their measures?

d. What kinds of angles are BAC and ECA? What does that ∠ ∠mean about their measures?

e. 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 2The figure below shows parallel lines 𝐿1 and 𝐿2. Let and be 𝑚 𝑛transversals that intersect 𝐿1 at points and , respectively, 𝐵 𝐶and 𝐿2 at point , as shown. Let be a point on 𝐹 𝐴 𝐿1 to the left of , be a point on 𝐵 𝐷 𝐿1 to the right of , be a point on 𝐶 𝐺 𝐿2 to the left of , and be a point on 𝐹 𝐸 𝐿2 to the right of .𝐹


EXPLORATORY CHALLENGE 2 - questions

a. Name the triangle in the figure.

b. Name a straight angle that will be useful in proving that the sum of the interior angles of the triangle is 180°.

c. Write your proof below.


Lesson Summary

MORE ON ANGLES OF A TRIANGLE

Lesson fourteen


NYS COMMON CORE MATHEMATICS CURRICULUM 8.2Lesson fourteen

DISCUSSION


1. Name an exterior angle and the related remote interior angles.2. Name a second exterior angle and the related remote interior angles.3. Name a third exterior angle and the related remote interior angles.4. Show that the measure of an exterior angle is equal to the sum of the

related remote interior angles.

EXERCISES 1-4


EXAMPLE ONE

Find the measure of angle . 𝑥


EXAMPLE TWO



EXAMPLE THREE



EXAMPLE FOUR



EXERCISE FIVE

Find the measure of angle . Present an informal 𝒙argument showing that your answer is correct.


EXERCISE SIX



EXERCISE SEVEN



EXERCISE EIGHT



EXERCISE NINE



EXERCISE TEN



Lesson Summary

Grade 8 The Concept of Congruence Module two. TOPIC C Congruence and Angle Relationships.

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Transcript of Grade 8 The Concept of Congruence Module two. TOPIC C Congruence and Angle Relationships.