Congruence of Triangles

Proving if some shapes, especially triangles, are congruent is an important part of the study of geometry. This lecture series discusses the three theorems that establish the congruence of triangles.

Practice

Prove triangle congruence - Questions

1. Holden tried to prove that $\triangle F G H \cong \triangle F I H$ .

	Statement	Reason
1	$\overline{I H} \cong \overline{G H}$	Given
2	$\overline{F H} \cong \overline{F H}$	Line segments are congruent to themselves.
3	$\triangle F G H \cong \triangle F I H$	Side-side congruence

What is the first error Holden made in his proof?

Choose 1 answer:

(A) Holden used an invalid reason to justify the congruence of a pair of sides or angles.

(B) Holden only established some of the necessary conditions for a congruence criterion.

(D) Holden used a criterion that does not guarantee congruence.

2. Prove that $\triangle P Q R \cong \triangle S T R$ .

	Statement	Reason
1	$\overline{Q R} \cong \overline{T R}$	Given
2	$\overline{P R} \cong \overline{S R}$	Given
3	Pick statement $(\angle P Q R \cong \angle S T R / \angle R P Q \cong \angle R S T / \angle Q R P \cong \angle T R S)$	Pick congruence criterion (Angle-angle-side/Angle-side-angle/Side-angle-side/Side-side-side)
4	$\triangle P Q R \cong \triangle S T R$	Pick reason (Given/Vertical angles are congruent./Corresponding parts of congruent triangles are congruent).

3. Prove that $\triangle A B C \cong \triangle A D C$ .

	Statement	Reason
1	$m \angle B A C=m \angle D A C=51^{\circ}$	Pick statement (Given / Vertical angles are congruent. / Linear pair angles are supplementary).
2	$A B=A D=4$	Given
3	Pick statement $( A B = A B / A C = A C / C D = C D )$	Segments are the same length as themselves.
4	$\triangle A B C \cong \triangle A D C$	Pick congruence criterion (angle-angle-side / Angle-side-angle / Side-angle-side / Side-side-side)

4. Jordy tried to prove that $\triangle A B E \cong \triangle B C D$ .

	Statement	Reason
1	$\angle B C D \cong \angle A B E$	Given
2	$\angle C D B \cong \angle B E A$	Given
3	$\overleftarrow{B D} \\| \overleftrightarrow{A E}$	Given
4	$\angle C B D \cong \angle B A E$	Corresponding angles on parallel lines are congruent.
5	$\triangle A B E \cong \triangle B C D$	Angle-angle-angle congruence

What is the first error Jordy made in his proof?

Choose 1 answer:

(A) Jordy used an invalid reason to justify the congruence of a pair of sides or angles.

(B) Jordy only established some of the necessary conditions for a congruence criterion.

(D) Jordy used a criterion that does not guarantee congruence.