Question

3&4. using the diagrams, is there enough information given to prove that the triangles are congruent? if so, state the theorem you would use

5&6. name the third congruence statement that is needed to prove that ABC is congruent to XYZ using the given theorem

Answer 1

There is enough information to prove that ΔABC and ΔDEF are congruent using the Angle-Side-Angle (ASA) congruency theorem

More information is required to prove that ΔGFH and ΔJKL are congruent

∠B ≅ ∠Y

∠A ≅ ∠X

In triangle ΔABC, and triangle ΔDEF, we have;

• Corresponding angles ∠A and ∠D are congruent
• Corresponding angles ∠C and ∠F are congruent
• Corresponding side in between the given angles ∠A and ∠C is congruent to the side in between the given angles ∠D and ∠F

Therefore ΔABC is congruent to ΔDEF by the Angle-Side-Angle (ASA) rule of congruency.

The corresponding angles ∠F and ∠J in triangle ΔGFH and triangle ΔJKL are congruent as well as the corresponding adjacent sides to the given angles and the corresponding opposite sides to the given angles are congruent, however, there is not enough information to prove that the two triangles are congruent.

The included angles, ∠H and ∠L between the sides congruent corresponding sides should be congruent or the other adjacent sides to the congruent angles ∠F and ∠J should be congruent for the triangles ΔGFH and ΔJKL to be congruent.

Given ∠B ≅ ∠Y, ∠C ≅ ∠Z, using the Angle-Angle-Side (AAS) rule of congruency, we have;

• The angles adjacent to ∠C and ∠Z, and opposite to ∠B and ∠Y which are ∠B and ∠Y are congruent

Therefore, we have ∠B ≅ ∠Y using AAS rule of congruency.

Using Angle-Side-Angle (ASA) rule, given ∠B ≅ ∠Y, ∠C ≅ ∠Z, then the angles on the other side of ∠B and ∠Y which are ∠A and ∠X should be congruent, we have;

• ∠A ≅ ∠X

Answer 2

3. There is enough information to prove that ΔABC and ΔDEF are congruent using the Angle-Side-Angle (ASA) congruency theorem

4. More information is required to prove that ΔGFH and ΔJKL are congruent

5. ∠B ≅ ∠Y

6. ∠A ≅ ∠X

Explanation:

3. In triangle ΔABC, and triangle ΔDEF, we have;

Corresponding angles ∠A and ∠D are congruent

Corresponding angles ∠C and ∠F are congruent

Corresponding side in between the given angles
`\overline{AC}` and
`\overline{DF}` are congruent

Therefore ΔABC is congruent to ΔDEF by the Angle-Side-Angle (ASA) rule of congruency

4. The corresponding angles ∠F and ∠J in triangle ΔGFH and triangle ΔJKL are congruent as well as the corresponding adjacent sides to the given angles and the corresponding opposite sides to the given angles are congruent, however, there is not enough information to prove that the two triangles are congruent

The included angles, ∠H and ∠L between the sides congruent corresponding sides should be congruent or the other adjacent sides
`\overline{FG}` and
`\overline{JK}` to the congruent angles ∠F and ∠J should be congruent for the triangles ΔGFH and ΔJKL to be congruent

5. Given
`\overline{AC}` ≅
`\overline{XZ}`, ∠C ≅ ∠Z, using the Angle-Angle-Side, AAS, rule of congruency, we have;

The angles adjacent to ∠C and ∠Z, and opposite to
`\overline{AC}` and
`\overline{XZ}`, which are B and ∠Y are congruent

Therefore, we have;

∠B ≅ ∠Y using AAS rule of congruency

6. Using Angle-Side-Angle, ASA, rule, given;

`\overline{AC}` ≅
`\overline{XZ}`, ∠C ≅ ∠Z, then the angles on the other side of
`\overline{AC}` and
`\overline{XZ}` which are ∠A and ∠X should be congruent, we have;

∠A ≅ ∠X