Question

1. (6 pts) in the diagram below, it is given that $overline{ad}congoverline{bc}$ and $angle adccongangle bcd$. prove that $angle dabcongangle abc$. 2. (6 pts) provide a new proof of itt part 1, completely different from what we did in the notes, by using the proof technique we saw in the proof of itt part 2. 3. (6 pts) provide a new proof of itt part 2, completely different from what we did in the notes, by using the proof technique we saw in the proof of itt part 1. 4. (8 pts) $\triangle xyz$ is given. prove, without using any triangle congruence: $overline{xy}congoverline{yz}congoverline{xz}$ if and only if $angle xcongangle ycongangle z$ note: this is a biconditional statement, so dont forget there are two separate “directions” to prove. 5. (7 pts) prove that the median from the vertex of an isosceles triangle is also an angle bisector and an altitude.

Explanation:

Step1: Prove \(\triangle ADC\cong\triangle BCD\)

We know that \(\overline{AD}\cong\overline{BC}\), \(\angle ADC\cong\angle BCD\), and \(\overline{DC}=\overline{DC}\) (common - side). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle ADC\cong\triangle BCD\).

Step2: Use congruent - triangle properties

Since \(\triangle ADC\cong\triangle BCD\), then \(\angle ACD\cong\angle BDC\) (corresponding parts of congruent triangles are congruent), and \(\overline{AC}\cong\overline{BD}\) (corresponding parts of congruent triangles are congruent).

Step3: Prove \(\triangle ABD\cong\triangle BAC\)

We have \(\overline{AD}\cong\overline{BC}\), \(\overline{AB}=\overline{AB}\) (common - side), and \(\overline{BD}\cong\overline{AC}\). By the Side - Side - Side (SSS) congruence criterion, \(\triangle ABD\cong\triangle BAC\).

Step4: Conclude the proof

Since \(\triangle ABD\cong\triangle BAC\), then \(\angle DAB\cong\angle ABC\) (corresponding parts of congruent triangles are congruent).

Answer:

We have proven that \(\angle DAB\cong\angle ABC\) using the SAS and SSS congruence criteria for triangles.

(Note: As the other sub - questions are also proof - based geometry problems, the general approach for each would be to identify relevant geometric properties, congruence criteria, and logical steps to reach the required conclusion. For example, for question 4, when proving the biconditional statement about an equilateral triangle, we would need to prove both "if \(\overline{XY}\cong\overline{YZ}\cong\overline{XZ}\) then \(\angle X\cong\angle Y\cong\angle Z\)" and "if \(\angle X\cong\angle Y\cong\angle Z\) then \(\overline{XY}\cong\overline{YZ}\cong\overline{XZ}\)" using angle - side relationships in triangles. For question 5, we would use the properties of isosceles triangles, medians, angle bisectors, and altitudes to prove the required result.)