1) if $ \angle a b c \cong \angle x y z $, then $ \overline{a c} \cong $ ?
a. $ \overline{x z} $
b. $ \overline{x y} $
c. $ \overline{y z} $
d. $ \overline{a b} $
2) if $ \angle c a t \cong \angle d o g $ then which of the following must be true?
a. $ \angle t a c \cong \angle d g o $
b. $ \overline{a t} \cong \overline{d o} $
c. $ \angle a c t \cong \angle o d g $
d. $ \overline{c t} \cong \overline{o g} $
3) if $ \triangle g e o \cong \triangle f u n $, find $ m \angle n $.
a $ 81^{circ} $
b. $ 38^{circ} $
c $ 180^{circ} $
d. $ 61^{circ} $
4) find $ m \angle b $.
a $ 23^{circ} $
b. $ 67^{circ} $
c $ 134^{circ} $
d. $ 46^{circ} $
5) given the triangle to the right, classify it by its sides.
a. scalene
b. isosceles
c. equilateral
d. acute
6) to prove the triangles congruent by asa, what other piece of information would i need?
a. $ \overline{b d} \cong \overline{d c} $
b. $ \angle b \cong \angle c $
c. $ \angle b d a \cong \angle c d a $
d. $ \angle b a d \cong \angle c a d $

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Accepted Answer

# Explanation:
## 1)
When $\triangle ABC\cong\triangle XYZ$, corresponding parts are congruent. By the definition of congruent triangles, $\overline{AC}$ corresponds to $\overline{XZ}$.

## 2)
If $\triangle CAT\cong\triangle DOG$, then $\angle TAC$ corresponds to $\angle ODG$, $\overline{AT}$ corresponds to $\overline{OG}$, $\angle ACT$ corresponds to $\angle DOG$, and $\overline{CT}$ corresponds to $\overline{DG}$.

## 3)
If $\triangle GEO\cong\triangle FUN$, then $\angle E$ corresponds to $\angle U = 38^{\circ}$, $\angle G=27^{\circ}$ corresponds to $\angle F$. Using the angle - sum property of a triangle ($\angle E+\angle G+\angle O = 180^{\circ}$ and $\angle U+\angle F+\angle N=180^{\circ}$), we know that $\angle O$ corresponds to $\angle N$. $\angle O=180^{\circ}-(27^{\circ} + 38^{\circ})=115^{\circ}$ (incorrect approach). Wait, no, if $\triangle GEO\cong\triangle FUN$, then $\angle O$ corresponds to $\angle N$. $\angle E = 38^{\circ}$, $\angle G = 81^{\circ}$, so $\angle O=180-(81 + 38)=61^{\circ}$ (using $\angle E+\angle G+\angle O = 180^{\circ}$ for $\triangle GEO$ and since $\triangle GEO\cong\triangle FUN$, $\angle N=\angle O$)

## 4)
In an isosceles triangle (sides $AB = AC$), the base - angles are equal. Using the angle - sum property of a triangle ($\angle A+\angle B+\angle C = 180^{\circ}$), and $\angle B=\angle C$. Given $\angle A = 46^{\circ}$, then $2\angle B=180 - 46$, $\angle B=\frac{180 - 46}{2}=67^{\circ}$

## 5)
A scalene triangle has all sides of different lengths. An isosceles triangle has at least two sides equal. An equilateral triangle has all sides equal. An acute triangle is classified by angles (all angles $<90^{\circ}$). Since the side - lengths are not given, but if we assume based on the problem's nature (if we consider the general case of triangle side - classification from the given options and no equal - side marks in the problem's description of the triangle for question 5), but wait, no, for a triangle with angles $30^{\circ},100^{\circ},50^{\circ}$,  side lengths are proportional to the sine of the angles. Since all angles are different, all side lengths are different (by the sine rule $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$), so it is scalene.

## 6)
For ASA (Angle - Side - Angle) congruence criterion. We know $\angle ADB=\angle ADC = 90^{\circ}$ (right angle) and $AD$ is a common side. To prove $\triangle ABD\cong\triangle ACD$ by ASA, we need $\angle BAD\cong\angle CAD$

# Answer:
1. a. $\overline{XZ}$
2. c. $\angle ACT\cong\angle DOG$
3. d. $61^{\circ}$
4. b. $67^{\circ}$
5. a. Scalene
6. d. $\angle BAD\cong\angle CAD$

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