use the figure below to answer the following questions.
11. if $\triangle lmn$ is isosceles and $t$ is the midpoint of $\overline{ln}$, which postulate can be used to prove $\triangle mlt \cong \triangle mnt$?
12. if $\triangle mlt \cong \triangle mnt$, what is used to prove $\angle 1 \cong \angle 2$
13. classify the triangle with coordinates $a(5, 0)$, $b(0, 5)$, and $c(-5, 0)$.
use the figure below to answer the following questions. find the measure of each angle.
14. $m\angle 1$
15. $m\angle 2$
16. $m\angle 3$
17. identify the transformation and verify that it is a congruence transformation.

Question

Accepted Answer

### Question 11
# Explanation:
## Step1: Analyze given information
$	riangle LMN$ is isosceles, so $ML = MN$ (legs of isosceles triangle). $T$ is the midpoint of $\overline{LN}$, so $LT = NT$. Also, $\overline{MT}$ is common to both $	riangle MLT$ and $	riangle MNT$.

## Step2: Identify congruence postulate
We have three sides: $ML = MN$, $LT = NT$, and $MT = MT$. So the SSS (Side - Side - Side) postulate can be used to prove $	riangle MLT\cong	riangle MNT$.
# Answer: SSS (Side - Side - Side) postulate

### Question 12
# Explanation:
## Step1: Recall properties of congruent triangles
If $	riangle MLT\cong	riangle MNT$, then corresponding parts of congruent triangles are congruent (CPCTC). $\angle1$ and $\angle2$ are corresponding angles of the congruent triangles $	riangle MLT$ and $	riangle MNT$.

## Step2: Apply CPCTC
By the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, since $	riangle MLT\cong	riangle MNT$, $\angle1\cong\angle2$.
# Answer: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

### Question 13
# Explanation:
## Step1: Calculate the lengths of the sides
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

- For $AB$: $A(5,0)$ and $B(0,5)$
$AB=\sqrt{(0 - 5)^2+(5 - 0)^2}=\sqrt{(- 5)^2+5^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}$

- For $BC$: $B(0,5)$ and $C(-5,0)$
$BC=\sqrt{(-5 - 0)^2+(0 - 5)^2}=\sqrt{(-5)^2+(-5)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}$

- For $AC$: $A(5,0)$ and $C(-5,0)$
$AC=\sqrt{(-5 - 5)^2+(0 - 0)^2}=\sqrt{(-10)^2+0^2}=\sqrt{100}=10$

## Step2: Classify the triangle
Since $AB = BC=5\sqrt{2}$ and $AC = 10$, and we can check the Pythagorean theorem: $(5\sqrt{2})^2+(5\sqrt{2})^2=50 + 50 = 100=10^2$. So it is an isosceles right triangle (two sides equal and satisfies Pythagorean theorem for right triangle).
# Answer: Isosceles right triangle

### Question 14 (m$\angle1$)
# Explanation:
## Step1: Identify the supplementary angle
The angle of $110^{\circ}$ and the adjacent angle (let's call it $\angle x$) on the straight line are supplementary. So $\angle x=180^{\circ}- 110^{\circ}=70^{\circ}$.

## Step2: Use the isosceles triangle property
The triangle with angle $\angle1$ has two equal sides (marked with ticks), so it is isosceles. The sum of angles in a triangle is $180^{\circ}$. Let $\angle1$ be $x$. Then $x + x+70^{\circ}=180^{\circ}$ (since the triangle is isosceles, the two base angles are equal).

## Step3: Solve for $\angle1$
$2x=180^{\circ}- 70^{\circ}=110^{\circ}$
$x = 55^{\circ}$
So $m\angle1 = 55^{\circ}$
# Answer: $55^{\circ}$

### Question 15 (m$\angle2$)
# Explanation:
## Step1: Identify the vertical angle
$\angle2$ and the angle of $70^{\circ}$ (from question 14) are vertical angles. Vertical angles are equal.

## Step2: Determine the measure of $\angle2$
Since vertical angles are congruent, $m\angle2=70^{\circ}$
# Answer: $70^{\circ}$

### Question 16 (m$\angle3$)
# Explanation:
## Step1: Analyze the right triangle
The triangle with $\angle3$ is a right triangle (marked with a right angle symbol). The sum of angles in a triangle is $180^{\circ}$. We know one angle is $90^{\circ}$ and another angle is equal to $\angle2 = 70^{\circ}$ (corresponding angles or alternate interior angles, since the lines are parallel in the figure).

## Step2: Calculate $\angle3$
$m\angle3=180^{\circ}-90^{\circ}-70^{\circ}=20^{\circ}$
# Answer: $20^{\circ}$

### Question 17
# Explanation:
## Step1: Identify the transformation
Looking at the grid, the transformation appears to be a reflection over the $x$-axis (since the $y$-coordinates of the points are negated while $x$-coordinates remain the same).

## Step2: Verify congruence transformation
A reflection is a congruence transformation because it preserves the distance between points (length of sides) and the measure of angles. For any two points $P(x,y)$ and $Q(a,b)$ in the original figure, the distance between them is $d=\sqrt{(a - x)^2+(b - y)^2}$. After reflection over the $x$-axis, the points become $P'(x,-y)$ and $Q'(a,-b)$. The distance between $P'$ and $Q'$ is $d'=\sqrt{(a - x)^2+(-b + y)^2}=\sqrt{(a - x)^2+(y - b)^2}=\sqrt{(a - x)^2+(b - y)^2}=d$. Also, the angles are preserved as reflection is a rigid motion.
# Answer: The transformation is a reflection (over the $x$-axis). It is a congruence transformation because it preserves side lengths and angle measures (rigid motion).

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QUESTION IMAGE

Question

Explanation:

Question 11

Step1: Analyze given information

Step2: Identify congruence postulate

Step1: Recall properties of congruent triangles

Step2: Apply CPCTC

Step1: Calculate the lengths of the sides

Step2: Classify the triangle

Answer:

Question 12