14. (40) in the figure at right, the measure of ∠d is 35° and the measure of ∠cab is 35°. find the measure of
a. ∠acb.
b. ∠acd.
c. ∠cad.
* 15. (inv. 3, 58) the y - axis is a line of symmetry for a triangle. the coordinates of two of its vertices are (-3, 2) and (0, 5).
a. what are the coordinates of the third vertex?
b. what is the area of the triangle?
* 16. (58)
a. a regular pentagon has how many lines of symmetry?
b. justify does a regular pentagon have rotational symmetry? how do you know?

Question

Accepted Answer

### Problem 14
#### Part a: Find $ \angle ACB $
# Explanation:
## Step1: Identify triangle type
Triangle $ ABC $ is right - angled at $ B $, so $ \angle B = 90^{\circ} $, and $ \angle CAB=35^{\circ} $.
## Step2: Use angle sum property
In $ \triangle ABC $, the sum of angles in a triangle is $ 180^{\circ} $. So $ \angle ACB=180^{\circ}-\angle B - \angle CAB $.
Substitute $ \angle B = 90^{\circ} $ and $ \angle CAB = 35^{\circ} $: $ \angle ACB=180 - 90-35=55^{\circ} $.
#### Part b: Find $ \angle ACD $
# Explanation:
## Step1: Identify linear pair
$ \angle ACB $ and $ \angle ACD $ form a linear pair, so $ \angle ACB+\angle ACD = 180^{\circ} $.
## Step2: Solve for $ \angle ACD $
We know $ \angle ACB = 55^{\circ} $, so $ \angle ACD=180 - 55 = 125^{\circ} $.
#### Part c: Find $ \angle CAD $
# Explanation:
## Step1: Identify triangle $ ACD $
In $ \triangle ACD $, $ \angle D = 35^{\circ} $, $ \angle ACD = 125^{\circ} $.
## Step2: Use angle sum property
Sum of angles in a triangle is $ 180^{\circ} $. So $ \angle CAD=180^{\circ}-\angle D-\angle ACD $.
Substitute $ \angle D = 35^{\circ} $ and $ \angle ACD = 125^{\circ} $: $ \angle CAD=180-(35 + 125)=20^{\circ} $.

### Problem 15
#### Part a: Coordinates of third vertex
# Explanation:
## Step1: Recall symmetry about y - axis
If a figure is symmetric about the $ y $ - axis, then for a point $ (x,y) $, its mirror image is $ (-x,y) $.
One vertex is $ (- 3,2) $, so its mirror image about $ y $ - axis is $ (3,2) $. The vertex $ (0,5) $ lies on the $ y $ - axis, so the third vertex is $ (3,2) $.
#### Part b: Area of the triangle
# Explanation:
## Step1: Identify base and height
The base of the triangle can be the distance between $ (-3,2) $ and $ (3,2) $. The distance between two points $ (x_1,y_1) $ and $ (x_2,y_2) $ with $ y_1 = y_2 $ is $ |x_2 - x_1| $. So base $ b=|3-(-3)| = 6 $.
The height is the vertical distance from the base (which is on the line $ y = 2 $) to the point $ (0,5) $, so height $ h=|5 - 2|=3 $.
## Step2: Use area formula for triangle
The area of a triangle is $ A=\frac{1}{2}\times base\times height $. Substitute $ b = 6 $ and $ h = 3 $: $ A=\frac{1}{2}\times6\times3 = 9 $.

### Problem 16
#### Part a: Lines of symmetry of regular pentagon
# Explanation:
## Step1: Recall regular polygon symmetry
A regular $ n $ - sided polygon has $ n $ lines of symmetry. For a regular pentagon, $ n = 5 $, so it has 5 lines of symmetry.
#### Part b: Rotational symmetry of regular pentagon
# Explanation:
## Step1: Recall rotational symmetry
A figure has rotational symmetry if it can be rotated by an angle $ \theta=\frac{360^{\circ}}{n} $ (where $ n $ is the number of sides for a regular polygon) and map onto itself.
For a regular pentagon, $ n = 5 $, so the angle of rotation is $ \frac{360^{\circ}}{5}=72^{\circ} $. If we rotate the regular pentagon by $ 72^{\circ} $, $ 144^{\circ} $, $ 216^{\circ} $, $ 288^{\circ} $ about its center, it maps onto itself. So a regular pentagon has rotational symmetry.

### Answers

#### Problem 14
a. $ \angle ACB=\boldsymbol{55^{\circ}} $
b. $ \angle ACD=\boldsymbol{125^{\circ}} $
c. $ \angle CAD=\boldsymbol{20^{\circ}} $

#### Problem 15
a. The third vertex is $ \boldsymbol{(3,2)} $
b. The area of the triangle is $ \boldsymbol{9} $ square units.

#### Problem 16
a. A regular pentagon has $ \boldsymbol{5} $ lines of symmetry.
b. Yes, a regular pentagon has rotational symmetry. Because when it is rotated by $ \frac{360^{\circ}}{5} = 72^{\circ} $ (and multiples of $ 72^{\circ} $ less than $ 360^{\circ} $) about its center, it maps onto itself.

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QUESTION IMAGE

Question

Explanation:

Problem 14

Part a: Find \( \angle ACB \)

Step1: Identify triangle type

Step2: Use angle sum property

Part b: Find \( \angle ACD \)

Step1: Identify linear pair

Step2: Solve for \( \angle ACD \)

Part c: Find \( \angle CAD \)

Step1: Identify triangle \( ACD \)

Step2: Use angle sum property

Problem 15

Part a: Coordinates of third vertex

Answer:

Problem 14

Problem 15

Problem 16