use the diagram and proof below to answer questions 10-11.

statement | reason
--- | ---
$\overline{jk} \cong \overline{dk}$ and $\overline{ae}$ bisects $\overline{hc}$ | 1) given
2) $\overline{hk} \cong \overline{ck}$ | 2) ______ (#10.)
3) ______ (#11.) | 3) vertical angles are congruent
4) $\triangle cdk \cong hjk$ | 4) side-angle-side

***#10.) which reason completes line 2 of the proof above?
(a) angle bisector definition
(b) midpoint definition
(c) segment bisector definition
(d) isosceles triangle definition

#11.) which statement completes line 3 of the proof above?
(a) $\angle kjh \cong \angle kdc$
(b) $\angle khj \cong \angle kcd$
(c) $\angle jkh \cong \angle dkc$
(d) $\angle fke \cong \angle bka$

**#12.) $\triangle mno$ is an isosceles triangle with base angles $\angle n$ and $\angle o$. given that $\overline{mo} = 5y + 7$ in., $\overline{mn} = 4y + 12$ in., and $\overline{on} = 2y + 10$ in., determine the perimeter of the triangle.
(a) 40 in
(b) 84 in
(c) 18 in.
(d) 21 in.

Question

use the diagram and proof below to answer questions 10-11.

statement | reason
--- | ---
$\overline{jk} \cong \overline{dk}$ and $\overline{ae}$ bisects $\overline{hc}$ | 1) given
2) $\overline{hk} \cong \overline{ck}$ | 2) ______ (#10.)
3) ______ (#11.) | 3) vertical angles are congruent
4) $\triangle cdk \cong hjk$ | 4) side-angle-side

***#10.) which reason completes line 2 of the proof above?
(a) angle bisector definition
(b) midpoint definition
(c) segment bisector definition
(d) isosceles triangle definition

#11.) which statement completes line 3 of the proof above?
(a) $\angle kjh \cong \angle kdc$
(b) $\angle khj \cong \angle kcd$
(c) $\angle jkh \cong \angle dkc$
(d) $\angle fke \cong \angle bka$

**#12.) $\triangle mno$ is an isosceles triangle with base angles $\angle n$ and $\angle o$. given that $\overline{mo} = 5y + 7$ in., $\overline{mn} = 4y + 12$ in., and $\overline{on} = 2y + 10$ in., determine the perimeter of the triangle.
(a) 40 in
(b) 84 in
(c) 18 in.
(d) 21 in.

Accepted Answer

### Question 10
# Brief Explanations:
To determine the reason for $\overline{HK} \cong \overline{CK}$, we analyze the definitions:
- **Angle Bisector Definition**: Applies to angles, not segments. Eliminate (A).
- **Midpoint Definition**: A midpoint divides a segment into two congruent parts. But here, $\overline{AE}$ bisects $\overline{HC}$, so $K$ is the midpoint of $\overline{HC}$, making $\overline{HK} \cong \overline{CK}$. This matches.
- **Segment Bisector Definition**: Describes a line/segment bisecting another, not the congruence of sub - segments. Eliminate (C).
- **Isosceles Triangle Definition**: Relates to triangle sides/angles, not segment bisecting. Eliminate (D).

# Answer: B. Midpoint Definition

### Question 11
# Brief Explanations:
Vertical angles are formed by intersecting lines and are congruent. We need vertical angles related to the triangles $\triangle CDK$ and $\triangle HJK$.
- $\angle JKH$ and $\angle DKC$ are vertical angles (formed by intersecting lines $JK$ - $DK$ and $HK$ - $CK$).
- Option (A): $\angle KJH$ and $\angle KDC$ are not vertical angles.
- Option (B): $\angle KHJ$ and $\angle KCD$ are not vertical angles.
- Option (D): $\angle FKE$ and $\angle BKA$ are vertical angles but not related to the triangles in the proof.

# Answer: C. $\angle JKH \cong \angle DKC$

### Question 12
# Explanation:
## Step 1: Identify equal sides in isosceles triangle
In $\triangle MNO$, base angles are $\angle N$ and $\angle O$, so the equal sides (legs) are $\overline{MO}$ and $\overline{NO}$? Wait, no—wait, base angles are $\angle N$ and $\angle O$, so the legs are $\overline{MN}$ and $\overline{MO}$? Wait, no, in an isosceles triangle, the sides opposite equal angles are equal. Base angles $\angle N$ and $\angle O$ are opposite $\overline{MO}$ and $\overline{MN}$ respectively? Wait, no, let's correct: In $\triangle MNO$, base angles are $\angle N$ and $\angle O$, so the sides opposite them ($\overline{MO}$ and $\overline{MN}$) are equal. So $MO = MN$.

Given $MO=5y + 7$, $MN = 4y+12$. Set them equal:
$5y + 7=4y + 12$

## Step 2: Solve for $y$
Subtract $4y$ from both sides: $y+7 = 12$
Subtract 7 from both sides: $y = 5$

## Step 3: Find side lengths
- $MO=5(5)+7=25 + 7=32$? Wait, no, that can't be. Wait, maybe I mixed up the sides. Wait, base angles are $\angle N$ and $\angle O$, so the equal sides are $\overline{MN}$ and $\overline{ON}$? Wait, no, let's re - examine the triangle label. The triangle is labeled $M$, $N$, $O$ with base $MN$? Wait, the diagram shows $M$ and $N$ at the base, $O$ at the top. So base angles are $\angle M$ and $\angle N$? Wait, the problem says "base angles $\angle N$ and $\angle O$". So the legs are $MO$ and $NO$? Wait, no, in a triangle, the base angles are the two angles adjacent to the base. If base angles are $\angle N$ and $\angle O$, the base is $MN$, and the legs are $MO$ and $NO$. Wait, but then $MO = NO$. Wait, $MO = 5y+7$, $ON=2y + 10$. Let's try that.

Set $MO = ON$:
$5y+7=2y + 10$

Subtract $2y$ from both sides: $3y+7 = 10$
Subtract 7: $3y=3$
$y = 1$

Wait, that gives $MO=5(1)+7 = 12$, $ON=2(1)+10 = 12$, $MN=4(1)+12 = 16$. Perimeter would be $12 + 12+16=40$. Wait, but let's check again. Wait, maybe the equal sides are $MN$ and $MO$. Let's try $MN = MO$:

$4y + 12=5y+7$
Subtract $4y$: $12=y + 7$
Subtract 7: $y = 5$

Then $MO=5(5)+7 = 32$, $MN=4(5)+12 = 32$, $ON=2(5)+10 = 20$. Perimeter $32 + 32+20=84$. Wait, but the options include 84 (option B). Wait, maybe I misidentified the equal sides. Let's recall: In an isosceles triangle, the sides opposite the base angles are equal. Base angles are $\angle N$ and $\angle O$, so the sides opposite them are $\overline{MO}$ (opposite $\angle N$) and $\overline{MN}$ (opposite $\angle O$)? No, $\angle N$ is at vertex $N$, opposite side is $\overline{MO}$; $\angle O$ is at vertex $O$, opposite side is $\overline{MN}$. So if $\angle N=\angle O$, then $\overline{MO}=\overline{MN}$. So $MO = MN$:

$5y + 7=4y+12$
$y = 5$

Then $MO = 5(5)+7=32$, $MN=4(5)+12 = 32$, $ON=2(5)+10 = 20$. Perimeter $=32 + 32+20=84$ inches.

# Answer: B. 84 in

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QUESTION IMAGE

Question

Explanation:

Question 10

Step 1: Identify equal sides in isosceles triangle

Step 2: Solve for \(y\)

Step 3: Find side lengths

Answer:

Question 11