Question

three: 8 pts each (24 points for this page)
properties of isosceles triangles:
part a (2 points):
75°
x =
part b (6 points): fill in the two missing blanks of this proof.
given: (overline{vw}congoverline{xw}), (overline{vz}congoverline{xy})
prove: (\triangle wzvcong\triangle wyx)
statements reasons
(overline{vw}congoverline{xw}) given
base angle theorem (properties of isosceles triangles)
(overline{vz}congoverline{xy}) given
(\triangle wzvcong\triangle wyx)
solve for x. you must show your set - up along with all work in order to receive full credit.
16.)
7x
10x + 10
x=
17.)
15x - 5
9 + 13x
x=

Explanation:

Part A

Step1: Recall angle - sum property of a triangle

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In the given triangle, the exterior angle is \(75^{\circ}\) and the two non - adjacent interior angles are equal (since the triangle is isosceles). Let the unknown angle be \(x\). So \(x + x=75^{\circ}\).

Step2: Solve for \(x\)

Combining like terms gives \(2x = 75^{\circ}\), then \(x=\frac{75^{\circ}}{2}=37.5^{\circ}\).

Part B

1. Since \(\overline{VW}\cong\overline{XW}\), by the Base - Angle Theorem, \(\angle V\cong\angle X\).
2. We are given \(\overline{VZ}\cong\overline{XY}\) and \(\overline{VW}\cong\overline{XW}\).
3. By the Side - Angle - Side (SAS) congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the two triangles are congruent. In \(\triangle WZV\) and \(\triangle WYX\), we have \(\overline{VW}\cong\overline{XW}\), \(\angle V\cong\angle X\) (from the Base - Angle Theorem) and \(\overline{VZ}\cong\overline{XY}\). So the missing statements and reasons are:

• Statement: \(\angle V\cong\angle X\), Reason: Base Angle Theorem (Properties of Isosceles Triangles)
• Statement: \(\triangle WZV\cong\triangle WYX\), Reason: Side - Angle - Side (SAS) Congruence Postulate

For problem 16

Step1: Set up the equation

Since the two angles are vertical angles, they are equal. So \(7x=10x + 10\).

Step2: Rearrange the equation

Subtract \(7x\) from both sides: \(0 = 10x+10 - 7x\), which simplifies to \(0 = 3x + 10\). Then subtract 10 from both sides: \(- 10=3x\).

Step3: Solve for \(x\)

Divide both sides by 3: \(x=-\frac{10}{3}\)

For problem 17

Step1: Set up the equation

Since the two angles are vertical angles, they are equal. So \(15x−5=9 + 13x\).

Step2: Rearrange the equation

Subtract \(13x\) from both sides: \(15x-13x−5=9\), which simplifies to \(2x−5 = 9\). Then add 5 to both sides: \(2x=9 + 5\), so \(2x=14\).

Step3: Solve for \(x\)

Divide both sides by 2: \(x = 7\)

Answer:

Part A: \(x = 37.5^{\circ}\)
Part B:

• Statement: \(\angle V\cong\angle X\), Reason: Base Angle Theorem (Properties of Isosceles Triangles)
• Statement: \(\triangle WZV\cong\triangle WYX\), Reason: Side - Angle - Side (SAS) Congruence Postulate

1. \(x=-\frac{10}{3}\)
2. \(x = 7\)