Question

proving congruence (triangle congruence: sss and sas)
classwork
for exercises 24 – 26, consider $\triangle mgt$. (you may want to sketch it.)

1. what angle is included between $\overline{gm}$ and $\overline{mt}$?
2. which sides include $\angle t$?
3. what angle is included between $\overline{gt}$ and $\overline{mg}$?
4. what additional information must be true to prove the two triangles congruent by sas triangle congruence?

for exercises 28 – 36, if the triangles can be proven congruent, state which rule applies and write a congruence statement. if not, write not enough information.
28.
29.
30.
31.
32.
33.
34.
35.
36.
geometry progressive mathematics initiative® (pmi®) congruent triangles
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Explanation:

Step1: Identify included angles/sides

For 24-26: The included angle between two sides of a triangle is the angle whose vertices are the endpoints of the two sides. The included sides of an angle are the two sides that form the angle.

Step2: Recall SAS congruence rule

For 27: SAS requires two pairs of congruent sides and the congruent included angle between them. The triangles have two pairs of congruent sides, so the included angles $\angle B$ and $\angle G$ must be congruent.

Step3: Apply triangle congruence rules

For 28: All three pairs of corresponding sides are congruent, so SSS applies.
For 29: No congruent sides/angles are marked, so no proof is possible.
For 30: Two pairs of corresponding sides are congruent, and the included angles are congruent, so SAS applies.
For 31: Two pairs of corresponding sides are congruent, and the included angles are congruent, so SAS applies.
For 32: Only one pair of sides and one pair of vertical angles are congruent, which is insufficient.
For 33: All three pairs of corresponding sides are congruent, so SSS applies.
For 34: Only two pairs of sides are congruent (vertical angles are congruent but do not form an included angle for SAS), so insufficient.
For 35: Two pairs of corresponding sides are congruent, and the included right angles are congruent, so SAS applies.
For 36: Two pairs of corresponding sides are congruent, and the included right angles are congruent, so SAS applies.

Answer:

1. 24. $\angle M$
2. 25. $\overline{TM}$ and $\overline{TD}$ (assuming $D$ is the third vertex of $\triangle DGT$)
3. 26. $\angle G$
4. 27. $\angle B \cong \angle G$ (or the included angles between the marked congruent sides)
5. 28. SSS; $\triangle ABC \cong \triangle DEF$
6. 29. not enough information
7. 30. SAS; $\triangle KLM \cong \triangle PON$
8. 31. SAS; $\triangle WZY \cong \triangle XZY$
9. 32. not enough information
10. 33. SSS; $\triangle BAD \cong \triangle BCD$
11. 34. not enough information
12. 35. SAS; $\triangle JKM \cong \triangle LKM$
13. 36. SAS; $\triangle ONQ \cong \triangle RQN$