Problem 3 Finding Congruent Triangles

Are the triangles congruent? Justify your answer.

A B ¯ ≅ E D ¯ Given B C ¯ ≅ D C ¯ B C = 4 = D C A C ¯ ≅ E C ¯ A C = 6 = E C ∠ A ≅ ∠ E , ∠ B ≅ ∠ D Given ∠ B C A ≅ ∠ D C E Vertical angles are congruent . table with 5 rows and 3 columns , row1 column 1 , eh b bar , column 2 approximately equal to , e d bar , column 3 cap given , row2 column 1 , b c bar , column 2 approximately equal to , d c bar , column 3 b c equals 4 equals d c , row3 column 1 , eh c bar , column 2 approximately equal to , e c bar , column 3 eh c equals 6 equals e c , row4 column 1 , angle eh , column 2 approximately equal to angle e comma angle b approximately equal to angle d , column 3 cap given , row5 column 1 , angle b c eh , column 2 approximately equal to angle d c e , column 3 cap verticalanglesarecongruent . . , end table

Δ A B C ≅ Δ E D C cap delta eh b c approximately equal to cap delta e d c  by the definition of congruent triangles.

✓ Got It?

3. Is Δ A B D ≅ Δ C B D ? cap delta eh b d approximately equal to cap delta c b d question mark  Justify your answer.

   

Take note Theorem 4-1: Third Angles Theorem

Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

If…
∠ A ≅ ∠ D angle eh approximately equal to angle d  and ∠ B ≅ ∠ E angle b approximately equal to angle e

Then…
∠ C ≅ ∠ F angle c approximately equal to angle f

Proof

Proof of Theorem 4-1: Third Angles Theorem

Given: ∠ A ≅ ∠ D , ∠ B ≅ ∠ E angle eh approximately equal to angle d comma angle b approximately equal to angle e

Prove: ∠ C ≅ ∠ F angle c approximately equal to angle f

Statements	Reasons
1) ∠ A ≅ ∠ D , ∠ B ≅ ∠ E angle eh approximately equal to angle d comma angle b approximately equal to angle e	1) Given
2) m ∠ A = m ∠ D , m ∠ B = m ∠ E m angle , eh equals , m angle d comma m angle , b equals . m angle e	2) Def .  of  ≅ cap def . , of , approximately equal to
3) m ∠ A + m ∠ B + m ∠ C = 180 , m angle , eh plus , m angle , b plus , m angle . c equals 180 comma m ∠ D + m ∠ E + m ∠ F = 180 m angle , d plus , m angle , e plus , m angle . f equals 180	3) Δ  Angle -Sum Thm. cap delta . cap angleminuscap sumcap thm.
4) m ∠ A + m ∠ B + m ∠ C = m ∠ D + m ∠ E + m ∠ F m angle , eh plus , m angle , b plus , m angle , c equals , m angle , d plus , m angle , e plus . m angle f	4) Subst. Prop.
5) m ∠ D + m ∠ E + m ∠ C = m ∠ D + m ∠ E + m ∠ F m angle , d plus , m angle , e plus , m angle , c equals , m angle , d plus , m angle , e plus . m angle f	5) Subst. Prop.
6) m ∠ C = m ∠ F m angle , c equals . m angle f	6) Subtraction Prop. of = cap subtractioncap prop.of . equals
7) ∠ C ≅ ∠ F angle c approximately equal to angle f	7) Def .  of  ≅ cap def . , of , approximately equal to