Chapter 4 – Congruent Triangles

4.2 and 4.9– Classifying Triangles and Isosceles, and Equilateral Triangles.

Match the letter of the figure to the correct vocabulary word in Exercises 1–4.

1.right triangle______

2.obtuse triangle______

3.acute triangle______

4.equiangular triangle______

Classify each triangle by its angle measures as acute, equiangular, right, or obtuse. (Note: Give two classifications for Exercise 7.)

5.6.7.

For Exercises 8–10, fill in the blanks to complete each definition.

8.An isosceles triangle has ______congruent sides.

9.An ______triangle has three congruent sides.

10.A ______triangle has no congruent sides.

Classify each triangle by its side lengths as equilateral, isosceles, or scalene. (Note: Give two classifications in Exercise 13.)

11.12.13.

Isosceles Triangles

Remember…

Isosceles triangles are triangles with at least two congruent sides.

The two congruent sides are called legs.

The third side is the base.

The two angles at the base are called base angles.

Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent.

Converse is true!


Find the value of x.

1. 2. 3.

4. 5.

Corollary 4.3 Angle Relationships in Triangles

  • The interior is the set of all points inside the figure.
  • The exterior is the set of all points outside the figure.
  • An interior angle is formed by two sides of a triangle.
  • An exterior angle is formed by one side of the triangle and extension of an adjacent side. It forms alinear pair with an angle of the triangle.
  • Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.

Exterior Angles: Find each angle measure.

37.mB ______38.mPRS ______

39. In LMN, the measure of an exterior angle at N measures 99.
and . Find mL, mM, and mLNM.______

40. mE and mG ______41.mT and mV ______

42.In ABC and DEF, mA  mD and mB  mE. Find mF if an exterior
angle at A measures 107, mB  (5x  2) , and mC  (5x  5) .______

43.The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle.______

44. One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle? ______

45. The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?

46. The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?

47. Find mB48. Find m<ACD

49. Find mK and mJ50. Find m<P and m<T

Use the figure at the right for problems 1-3.

  1. Find m3 if m5 = 130 and m4 = 70.
  1. Find m1 if m5 = 142 and m4 = 65.
  1. Find m2 if m3 = 125 and m4 = 23.

Use the figure at the right for problems 4-7.

  1. m6 + m7 + m8 = ______.
  1. If m6 = x, m7 = x – 20, and m11 = 80,

then x = _____.

  1. If m8 = 4x, m7 = 30, and m9 = 6x -20,

then x = _____.

  1. m9 + m10 + m11 = ______.

For 8 – 12, solve for x.

8. 9.

4.4 Congruent Triangles

Polygons are congruent if all of their corresponding sides and all of their corresponding angles are congruent.

Consecutive vertices of a polygon– the endpoints of a side

Ex. P and Q are consecutive vertices

Opposite vertices of a polygon- vertices that are not consecutive

Congruent riangles: Two ’s are  if they can be matched up so that corresponding angles and sides of the ’s are .

Congruence Statement: A congruence statement matches up the parts in the same order.

RED  FOX

List the corresponding ’s:corresponding sides:

R  ___  ____

E  ___  ____

D  ___  ____

Examples:

1. The two ’s shown are .

a) ABO  _____b) A  ____

c)  _____d) BO = ____

2. The pentagons shown are .

a) B corresponds to ____b) BLACK  ______

c) ______= mEd) KB = ____ cm

e) If  , name two right ’s in the figures.

3. Given BIG  CAT, BI = 14, IG = 18, BG = 21, CT = 2x + 7. Find x.

The following ’s are , complete the congruence statement:

Parts of a Triangle in terms of their relative positions.

1. Name the opposite side to C.

2. Name the included side between A and B.

3. Name the opposite angle to .

4. Name the included angle between and .

4.5-4.7 Proving Triangles Congruent

Ways to Prove ’s :

SSS Postulate: (side-side-side) Three sides of one  are  to three sides of a second ,

Given: bisects ;

SAS Postulate: (side-angle-side) Two sides and the included angle of one  are  to two sides

and the included angle of another .

Given: bisects AXE;

ASA Postulate: (angle-side-angle) Two angles and the included side of one  are  to two angles

and the included side of another .

Given:

AAS Theorem: (angle-angle-side) Two angles and a non-included side of one  are  to two

angles and a non-included side of another .

Given:

HL Theorem: (hypotenuse-leg) The hypotenuse and leg of one right  are  to the hypotenuse

and leg of another right .

Given:

Isosceles  FAC with legs

CPCTC – Corresponding Parts of Congruent Triangles are Congruent

State which congruence method(s) can be used to prove the ∆s . If no method applies, write none. All markings must correspond to your answer.

1. 2.

3. 4.

5. 6.

7. 8.

State which congruence method(s) can be used to prove the ∆s . If no method applies, write none. All markings must correspond to your answer.

1. 2. 3.

4. 5. 6.

7. 8.9.

Fill in the congruence statement and then name the postulate that proves the ∆s are . If the ∆s are not , write “not possible” in second blank. (Leave first blank empty)*Markings must go along with your answer** Some may have more than one postulate

1. 2.

∆ABC  _____ by ______∆ABC  ______by ______

3. 4.

∆ABC  ______by ______∆ABC  ______by ______

5. 6.

∆ABC  ______by ______∆ABC  ______by ______

7. 8.

∆ABC  ______by ______∆ABC  ______by ______

9. 10.

∆ABC  ______by ______∆ABC  ______by ______

11. 12.

∆ABC  ______by ______∆ABC  ______by ______

13. 14.

∆ABC  ______by ______∆ABC  ______by ______

Proofs!

#1 Given: Prove:

1. 1. ______

2. 1  42. ______

3. ∆RST  ∆TUV3. ______

4. 3  24. ______

5. 5. ______

#2 Given: D is the midpoint of Prove: bisects ACB.

1. D is the midpoint of 1. ______

2. 2. ______

3. 3. ______

4. ∆ACD  ∆BCD4. ______

5. 1  25. ______

6. bisects ACB.6. ______

#3 Given: AR AQ; RS QT Prove: AS AT

1. AR AQ; RS QT1. ______

2. <R  <Q2. ______

3. ARS  AQT3. ______

4. AS AT 4. ______

#1

Given:

Prove: Δ ADB Δ CDB

1. 1. ______

2. 2. ______

3.  1 & 2 are right ’s.3. ______

4.  1 24. ______

5. 5. ______

6. Δ ADB Δ CDB6. ______

#2

Given:

bisects ADC

Prove:

1. 1. ______

2.  1 & 2 are right ’s2. ______

3. 1 23. ______

4. 4. ______

5. bisects ADC5. ______

6.  3 46. ______

7. Δ ADB Δ CDB7. ______

8. 8. ______

Congruent Triangles Proofs

1.Given: ; O is the midpoint of

Prove: O is the midpoint of

2.Given: ; D is the midpoint of

Prove:

3.Given:

Prove:

4.Given:

M is the midpoint

Prove:

5.Given:

B is the midpoint of

Prove:

6.Given:

Prove: bisects

1