Question

complete proofs 1 - 2 using angle - side - angle (asa).

1. given: $overline{bd}$ bisects $angle abc$, $angle bdacongangle bdc$

prove: $\triangle abdcong\triangle cbd$
statements reasons

1. $overline{bd}$ bisects $angle abc$ 1. given
2. $angle bdacongangle bdc$ 2. given
3. 3.
4. 4.
5. $\triangle abdcong\triangle cbd$ 5. asa
6. given: $overline{st}paralleloverline{uv}$, $overline{st}congoverline{vu}$

prove: $\triangle stwcong\triangle vuw$
statements reasons

1. 1.
2. 2.
3. 3.
4. 4.
5. 5.

complete proofs 3 - 5 using angle - angle - side (aas).

1. given: $angle kjlcongangle lmk$, $angle jklcongangle mlk$

prove: $\triangle jklcong\triangle mlk$
statements reasons

1. 1.
2. 2.
3. 3.
4. 4.

Explanation:

Step1: Identify common side

$BD = BD$ (Reflexive property of congruence)

Step2: Apply ASA

Since $\overline{BD}$ bisects $\angle ABC$, $\angle ABD=\angle CBD$ (Definition of angle - bisector). We are given $\angle BDA\cong\angle BDC$ and $BD = BD$. So, $\triangle ABD\cong\triangle CBD$ (ASA)

Step1: Use parallel - line property

Since $\overline{ST}\parallel\overline{UV}$, $\angle TSW=\angle VUW$ (Alternate interior angles)

Step2: Identify vertical angles

$\angle SWT=\angle UWV$ (Vertical angles are congruent)

Step3: Given side equality

We are given $\overline{ST}\cong\overline{VU}$

Step4: Apply ASA

So, $\triangle STW\cong\triangle VUW$ (ASA)

Step1: Identify common side

$KL = KL$ (Reflexive property of congruence)

Step2: Apply AAS

We are given $\angle KJL\cong\angle LMK$ and $\angle JKL\cong\angle MLK$, and $KL = KL$. So, $\triangle JKL\cong\triangle MLK$ (AAS)

Answer:

1. $BD = BD$, Reflexive property of congruence
2. $\angle ABD=\angle CBD$, Definition of angle - bisector

For the second problem: