Question

#1: △hey is congruent to △man by aas≅. what other parts of the triangles are congruent by cpctc?
#2: △cat≅△rap, by asa
therefore:
#3: given: $overline{ac}congoverline{ar}$ and $angle1congangle2$
prove: $angle3congangle4$
proof:

1. $overline{ac}congoverline{ar}$

2.

1. $angle calcongangle ras$
2. △lca≅△sra
3. $angle3congangle4$

Explanation:

Step1: Recall CPCTC (Corresponding - Parts of Congruent Triangles are Congruent)

If \(\triangle HEY\cong\triangle MAN\) by \(AAS\), then:
\(\overline{HE}\cong\overline{MA}\), \(\overline{EY}\cong\overline{AN}\), \(\overline{HY}\cong\overline{MN}\), \(\angle H\cong\angle M\), \(\angle E\cong\angle A\), \(\angle Y\cong\angle N\)

Step2: For \(\triangle CAT\cong\triangle RAP\) by \(ASA\)

By CPCTC, \(\overline{CA}\cong\overline{RA}\), \(\overline{AT}\cong\overline{AP}\), \(\overline{CT}\cong\overline{RP}\), \(\angle C\cong\angle R\), \(\angle T\cong\angle P\), \(\angle CAT\cong\angle RAP\)

Step3: Prove \(\angle3\cong\angle4\)

1. Given \(\overline{AC}\cong\overline{AR}\)
2. Given \(\angle1\cong\angle2\)
3. \(\angle CAL\) and \(\angle RAS\) are vertical - angles, so \(\angle CAL\cong\angle RAS\) (Vertical angles are congruent)
4. \(\triangle LCA\cong\triangle SRA\) by \(ASA\) (Since \(\overline{AC}\cong\overline{AR}\), \(\angle CAL\cong\angle RAS\), \(\angle1\cong\angle2\))
5. \(\angle3\cong\angle4\) by CPCTC

Answer:

#1: \(\overline{HE}\cong\overline{MA}\), \(\overline{EY}\cong\overline{AN}\), \(\overline{HY}\cong\overline{MN}\), \(\angle H\cong\angle M\), \(\angle E\cong\angle A\), \(\angle Y\cong\angle N\)
#2: \(\overline{CA}\cong\overline{RA}\), \(\overline{AT}\cong\overline{AP}\), \(\overline{CT}\cong\overline{RP}\), \(\angle C\cong\angle R\), \(\angle T\cong\angle P\), \(\angle CAT\cong\angle RAP\)
#3:

1. Given
2. Given
3. Vertical angles are congruent
4. \(\triangle LCA\cong\triangle SRA\) by \(ASA\)
5. CPCTC