Warm-Up Exercises

4.1 Apply Triangle Sum Properties 4.2 Apply Congruence and Triangles 4.3 Relate Transformations and Congruence 4.4 Prove Triangles Congruent by SSS 4.5 Prove Triangles Congruent by SAS and HL 4.6 Prove Triangles Congruent by ASA and AAS 4.7 Use Congruent Triangles 4.8 Use Isosceles and Equilateral Triangles 4.9 Perform Congruence Transformations

CHAPTER 4: Congruent Triangles

Warm-Up Exercises

•  classifying angles •  solving linear equations •  finding midpoints •  using angle relationships

In previous courses, you learned the following skills, which you’ll use in Chapter 4:

Warm-Up Exercises Prerequisite Skills: VOCABULARY CHECK Classify the angle as acute, obtuse, right, or straight.

1.m∠A = 115! Obtuse

2.m∠B = 90! Right

3.m∠C = 35! Acute

4.m∠A = 95! Obtuse

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

Solve the equation.

70 + 2y = 18070 − 70 + 2y = 180 − 702y = 1102y2

= 1102

= 55

5. 70 + 2y = 180

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

Solve the equation. 2x = 5x − 542x − 2x = 5x − 2x − 540 + 54 = 3x − 54 + 5454 = 3x543

= 3x3

x = 18

6. 2x = 5x − 54

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

Solve the equation.

40 + x + 65 = 180x +105 = 180x +105 −105 = 180 −105x = 75

7. 40 + x + 65 = 180

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

M = x1 + x22

, y1 + y22

⎛⎝⎜

⎞⎠⎟

M = 2 + (−1)2

, (−5)+ (−2)2

⎛⎝⎜

⎞⎠⎟

M = 12,− 72

⎛⎝⎜

⎞⎠⎟

8. P(2,−5),Q(−1,−2)Find the coordinates of the midpoint of PQ.

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

M = x1 + x22

, y1 + y22

⎛⎝⎜

⎞⎠⎟

M = −4 +12

, 7 + (−5)2

⎛⎝⎜

⎞⎠⎟

M = − 32,1⎛

⎝⎜⎞⎠⎟

9. P(−4,7),Q(1,−5)Find the coordinates of the midpoint of PQ.

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

M = x1 + x22

, y1 + y22

⎛⎝⎜

⎞⎠⎟

M = h + h2, k2

⎛⎝⎜

⎞⎠⎟

M = h, k2

⎛⎝⎜

⎞⎠⎟

10. P(h,k),Q(h,0)Find the coordinates of the midpoint of PQ.

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

Determine whether the angles are congruent. If so, explain why.

11. ∠2,∠3

Yes, Vertical Angles Congruence Theorem

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

Determine whether the angles are congruent. If so, explain why.

12. ∠1,∠4

Yes, Corresponding Angles Postulate

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

Determine whether the angles are congruent. If so, explain why.

13. ∠2,∠6

Yes, Alternate Interior Angles Theorem

Warm-Up Exercises Prerequisite Skills: SKILLS AND ALGEBRA CHECK

Determine whether the angles are congruent. If so, explain why.

14. ∠3,∠4

No

Warm-Up Exercises

! Classifying triangles by sides and angles ! Proving that triangles are congruent ! Using coordinate geometry to investigate triangle relationships

In Chapter 4, you will apply the BIG IDEAS listed below and reviewed in the Chapter Summary. You will also use the key vocabulary listed in the following slide.

Warm-Up Exercises

• triangle scalene, isosceles, equilateral, acute, right, obtuse, equiangular • interior angles • exterior angles • corollary • congruent figures • corresponding parts

In Chapter 4, you will also use the key vocabulary listed below.

Warm-Up Exercises

• right triangle legs, hypotenuse • flow proof • isosceles triangle legs, vertex angle, base, base angles • transformation translation, reflection, rotation

In Chapter 4, you will also use the key vocabulary listed below.

Warm-Up Exercises

1. 90º

ANSWER right

2. 72º

Classify each angle as acute, obtuse, or right.

ANSWER acute

Warm Up Exercises

Warm-Up Exercises

3. 116º

ANSWER obtuse

Classify each angle as acute, obtuse, or right.

4. How do you know that 1 = 2? ~

2

1 ANSWER Alt. Int. s Thm.

Warm Up Exercises

Warm-Up Exercises 4.1 Apply Triangle Sum Properties Goal: Classify triangles and find measures of their angles.

Key Vocabulary: •  triangle scalene, isosceles, equilateral, acute, right, obtuse, equiangular • interior angles • exterior angles • corollary to a theorem

Warm-Up Exercises 4.1 Apply Triangle Sum Properties

Goal: Classify triangles and find measures of their angles.

Postulates, Corollaries, and Theorems: •  Theorem 4.1: Triangle Sum Theorem •  Theorem 4.2: Exterior Angle Theorem •  Corollary to the Triangle Sum Theorem

Warm-Up Exercises Definitions TRIANGLE: A polygon with three sides.

Warm-Up Exercises Definitions SCALENE TRIANGLE: A triangle with no congruent sides.

Warm-Up Exercises Definitions ISOSCELES TRIANGLE: A triangle with at least two congruent sides.

Warm-Up Exercises Definitions EQUILATERAL TRIANGLE: A triangle with three congruent sides.

Warm-Up Exercises Definitions ACUTE TRIANGLE: A triangle with three acute angles.

Warm-Up Exercises Definitions RIGHT TRIANGLE: A triangle with one right angle.

Warm-Up Exercises Definitions OBTUSE TRIANGLE: A triangle with one obtuse angle.

Warm-Up Exercises Definitions EQUIANGULAR TRIANGLE: A triangle with three congruent angles.

Warm-Up Exercises Definitions INTERIOR ANGLES OF A TRIANGLE: When the sides of a triangle are extended, the three original angles of the triangle.

Warm-Up Exercises Definitions EXTERIOR ANGLES OF A TRIANGLE: When the sides of a triangle are extended, the angles that are adjacent to the interior angles. The exterior angles from linear pairs with the interior angles.

Warm-Up Exercises Definitions COROLLARY TO A THEOREM: A statement that can be proved easily using the theorem.

The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

Warm-Up Exercises EXAMPLE 1 Classify triangles by sides and by angles

SOLUTION

The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 55° , 55° , and 70° . It is an acute isosceles triangle.

Support Beams

Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.

Warm-Up Exercises EXAMPLE 2 Classify a triangle in a coordinate plane

SOLUTION

STEP 1 Use the distance formula to find the side lengths.

Classify PQO by its sides. Then determine if the triangle is a right triangle.

OP = y2 – y1 ( ) 2 x2 – x1 ( ) 2 +

= 2 – 0( ) 2 (– 1 ) 0( ) 2 + – = 5 2.2

OQ = y2 – y1 ( ) 2 x2 – x1 ( ) 2 + 2 = – 0( ) 6 0( ) 2 + – 3 = 45 6.7

Warm-Up Exercises EXAMPLE 2 Classify a triangle in a coordinate plane

PQ = y2 – y1 ( ) 2 x2 – x1 ( ) 2 +

3 – 2( ) 2 6 ( ) 2 + – = (– 1 ) = 50 7.1

STEP 2 Check for right angles.

The slope of OP is 2 – 0 – 2 – 0

= – 2.

The slope of OQ is 3 – 0 6 – 0

= 2 1 .

1 The product of the slopes is – 2 2 = – 1 ,

so OP OQ and POQ is a right angle.

Therefore, PQO is a right scalene triangle. ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2

1.  Draw an obtuse isosceles triangle and an acute scalene triangle.

obtuse isosceles triangle

B

A C

acute scalene triangle P

Q

R

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2

2.  Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its sides. Then determine if it is a right triangle.

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2

2.  Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its sides. Then determine if it is a right triangle.

ABC is a right Isosceles triangle.

ANSWER

Warm-Up Exercises THEOREM 4.1: Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°.

Warm-Up Exercises Prove the Triangle Sum Theorem

1. Parallel Postulate

2. Angle Addition Postulate and definition of straight angle

3. Alternate Interior Angles Thm

4. Definition of congruent angles

1. Draw BD! "##

parallel to AC.

GIVEN: ΔABC PROVE: m∠1+m∠2 +m∠3= 180!

2.m∠4 +m∠2 +m∠5 = 180!

3. ∠1≅ ∠4;∠3≅ ∠5

4. m∠1= m∠4;m∠3= m∠5

5. m∠1+m∠2 +m∠3= 180! 5. Substitution Property of Equality

Warm-Up Exercises THEOREM 4.2: Exterior Angle Theorem - The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Warm-Up Exercises Prove the Exterior Angle Theorem

1. Given

2. Triangle Sum Theorem

3. Transitive Property of Equality

4. Subtraction Property of Equality

1. m∠ACB +m∠BCD = 180!

GIVEN: ΔACBPROVE: m∠BCD = m∠A +m∠B

2. m∠A +m∠B +m∠ACB = 180!

3. m∠ACB +m∠BCD = m∠A +m∠B +m∠ACB

4. m∠BCD = m∠A +m∠B

Warm-Up Exercises EXAMPLE 3 Find an angle measure

SOLUTION

STEP 1 Write and solve an equation to find the value of x.

Apply the Exterior Angle Theorem. (2x – 5)° = 70° + x°

Solve for x. x = 75

STEP 2 Substitute 75 for x in 2x – 5 to find m∠ JKM.

2x – 5 = 2 75 – 5 = 145

Find m∠ JKM.

The measure of ∠ JKM is 145°. ANSWER

Warm-Up Exercises COROLLARY TO THE TRIANGLE SUM THEOREM: The acute angles of a right triangle are complementary.

Warm-Up Exercises Prove the Corollary to the Triangle Sum Theorem

1. Given

2. Definition of right angle

3. Triangle Sum Theorem

4. Substitution Property of Equality

1. ΔABC is a right triangle

GIVEN: ΔABC is a right trianglePROVE: ∠A and ∠B are complementary

2. m∠C = 90!

3. m∠A +m∠B +m∠C = 180!

4. m∠A +m∠B + 90! = 180!

5. m∠A +m∠B = 90! 5. Subtraction Property of Equality

6. ∠A and ∠B are complementary 6. Definition of complementary angles

Warm-Up Exercises EXAMPLE 4 Find angle measures from a verbal description

Use the corollary to set up and solve an equation.

Corollary to the Triangle Sum Theorem x° + 2x° = 90°

Solve for x. x = 30

So, the measures of the acute angles are 30° and 2(30°) = 60° .

ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4

Find the measure of 1 in the diagram shown. 3 .

The measure of ∠ 1 in the diagram is 65°. ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4

SOLUTION

A + B + C = 180°

x + 2x + 3x = 180°

6x = 180°

x = 30°

B = 2x = 2(30) = 60°

C = 3x = 3(30) = 90°

x

2x 3x

4 .Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°. °

Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4

5 .Find the measures of the acute angles of the right triangle in the diagram shown.

26° and 64° ANSWER

Warm-Up Exercises Daily Homework Quiz

1. Graph ABC with vertices A(0, 6), B(– 4, – 1) and C(4, – 1). Classify it by its sides. Then determine if it is a right triangle.

isosceles; not a right angle ANSWER

Warm-Up Exercises Daily Homework Quiz

2. Find x. Then classify the triangle by its angles.

22; acute ANSWER

3. Find the measure of exterior angle shown.

104° ANSWER

Warm-Up Exercises Daily Homework Quiz

4. Find x and y.

82, 58 ANSWER

Warm-Up Exercises Closing •  Equilateral triangles have three congruent sides, isosceles triangles have at least two congruent sides, and scalene triangles have no congruent sides. •  Equiangular triangles have three congruent angles, acute triangles have three acute angles, obtuse triangles have one obtuse angle, and right triangles have one right angle. •  The sum of the measures of the interior angles of a triangle is 180°.

Warm-Up Exercises Closing Add the two known angle measures and subtract the result from 180° to find the measure of the third angle.

Warm-Up Exercises 4.1 Homework

•  P211: 1-37 odd