Question

1. in △pbl, what is the included angle for (overline{bp}) and (overline{pl})?

a. (angle p) b. (angle b) c. (angle l) d. (overline{pb})

1. which parts must be congruent to prove that △mud ≅ △cat by the asa postulate?

a. (overline{mu}congoverline{ca}) b. (overline{md}congoverline{ct}) c. (overline{ud}congoverline{at}) d. (angle mcongangle c)

1. which postulate or theorem can be used to prove that △tlc ≅ △bkw?

a. asa b. ssa c. sas d. the triangles cannot be proved to be congruent.

1. which parts must be congruent to prove that △bul ≅ △dog by the hl theorem?

a. (angle lcongangle g) b. (overline{bl}congoverline{dg}) c. (overline{bu}congoverline{do}) d. none of these

Explanation:

Step1: Recall included - angle definition

The included angle between two sides of a triangle is the angle formed by those two sides. In \(\triangle PBL\), the included angle for \(\overline{BP}\) and \(\overline{PL}\) is \(\angle P\).

Step2: Recall ASA (Angle - Side - Angle) postulate

The ASA postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. For \(\triangle MUD\) and \(\triangle CAT\) to be congruent by ASA, we need \(\angle M\cong\angle C\), \(\overline{MU}\cong\overline{CA}\) and \(\angle U\cong\angle A\). The key part among the options is \(\angle M\cong\angle C\).

Step3: Analyze congruence postulates and given markings

Looking at \(\triangle TLC\) and \(\triangle BKW\), we have two sides and the included angle congruent (the markings show side - angle - side congruence). So, we use the SAS (Side - Angle - Side) postulate.

Step4: Recall HL (Hypotenuse - Leg) theorem

The HL theorem is used for right - triangles. It states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. For \(\triangle BUL\) and \(\triangle DOG\) to be congruent by HL, we need \(\overline{BL}\cong\overline{DG}\) (the hypotenuses).

Answer:

1. A. \(\angle P\)
2. D. \(\angle M\cong\angle C\)
3. C. SAS
4. B. \(\overline{BL}\cong\overline{DG}\)