Question

unit 2 quiz #1
show your work for full credit. read the instructions.
classify each △ by sides and angles.
①
sides:
angles:
②
sides:
angles:
③
sides:
angles:
④
sides:
angles:
find the missing angle(s) by using the triangle sum theorem.
⑤
⑥

Explanation:

Let's solve the missing angle problems (5, 6, and the right triangle at the bottom) using the Triangle Sum Theorem, which states that the sum of the interior angles of a triangle is \(180^\circ\).

Problem 5:

We have triangle \(ABC\) with \(\angle A = 70^\circ\), \(\angle B = 27^\circ\), and we need to find \(\angle C = x\).

Step 1: Recall the Triangle Sum Theorem

The sum of the angles in a triangle is \(180^\circ\), so \(\angle A + \angle B + \angle C = 180^\circ\).

Step 2: Substitute the known values

Substitute \(\angle A = 70^\circ\), \(\angle B = 27^\circ\), and \(\angle C = x\) into the equation:
\[
70^\circ + 27^\circ + x = 180^\circ
\]

Step 3: Simplify and solve for \(x\)

First, add \(70^\circ\) and \(27^\circ\):
\[
97^\circ + x = 180^\circ
\]
Then, subtract \(97^\circ\) from both sides:
\[
x = 180^\circ - 97^\circ = 83^\circ
\]

Problem 6:

We have an isosceles triangle (two equal angles, \(x^\circ\) each) with a third angle of \(54^\circ\).

Step 1: Recall the Triangle Sum Theorem

The sum of the angles in a triangle is \(180^\circ\), so \(x + x + 54^\circ = 180^\circ\).

Step 2: Combine like terms

Combine the two \(x\) terms:
\[
2x + 54^\circ = 180^\circ
\]

Step 3: Solve for \(x\)

Subtract \(54^\circ\) from both sides:
\[
2x = 180^\circ - 54^\circ = 126^\circ
\]
Then, divide both sides by 2:
\[
x = \frac{126^\circ}{2} = 63^\circ
\]

Right Triangle (Bottom):

We have a right triangle (one angle is \(90^\circ\)) with angles \((2x + 1)^\circ\), \((6x + 5)^\circ\), and \(90^\circ\).

Step 1: Recall the Triangle Sum Theorem

The sum of the angles in a triangle is \(180^\circ\), so \((2x + 1)^\circ + (6x + 5)^\circ + 90^\circ = 180^\circ\).

Step 2: Combine like terms

Combine the \(x\) terms and the constant terms:
\[
(2x + 6x) + (1 + 5) + 90^\circ = 180^\circ
\]
\[
8x + 6 + 90 = 180
\]
\[
8x + 96 = 180
\]

Step 3: Solve for \(x\)

Subtract 96 from both sides:
\[
8x = 180 - 96 = 84
\]
Then, divide both sides by 8:
\[
x = \frac{84}{8} = 10.5
\]

Now, we can find the angles:

• \((2x + 1)^\circ = 2(10.5) + 1 = 22^\circ\)
• \((6x + 5)^\circ = 6(10.5) + 5 = 68^\circ\)

Summary of Answers:

• Problem 5: \(x = 83^\circ\)
• Problem 6: \(x = 63^\circ\)
• Right Triangle: \(x = 10.5\), angles are \(22^\circ\), \(68^\circ\), and \(90^\circ\)

Answer:

Let's solve the missing angle problems (5, 6, and the right triangle at the bottom) using the Triangle Sum Theorem, which states that the sum of the interior angles of a triangle is \(180^\circ\).

Problem 5:

We have triangle \(ABC\) with \(\angle A = 70^\circ\), \(\angle B = 27^\circ\), and we need to find \(\angle C = x\).

Step 1: Recall the Triangle Sum Theorem

The sum of the angles in a triangle is \(180^\circ\), so \(\angle A + \angle B + \angle C = 180^\circ\).

Step 2: Substitute the known values

Substitute \(\angle A = 70^\circ\), \(\angle B = 27^\circ\), and \(\angle C = x\) into the equation:
\[
70^\circ + 27^\circ + x = 180^\circ
\]

Step 3: Simplify and solve for \(x\)

First, add \(70^\circ\) and \(27^\circ\):
\[
97^\circ + x = 180^\circ
\]
Then, subtract \(97^\circ\) from both sides:
\[
x = 180^\circ - 97^\circ = 83^\circ
\]

Problem 6:

We have an isosceles triangle (two equal angles, \(x^\circ\) each) with a third angle of \(54^\circ\).

Step 1: Recall the Triangle Sum Theorem

The sum of the angles in a triangle is \(180^\circ\), so \(x + x + 54^\circ = 180^\circ\).

Step 2: Combine like terms

Combine the two \(x\) terms:
\[
2x + 54^\circ = 180^\circ
\]

Step 3: Solve for \(x\)

Subtract \(54^\circ\) from both sides:
\[
2x = 180^\circ - 54^\circ = 126^\circ
\]
Then, divide both sides by 2:
\[
x = \frac{126^\circ}{2} = 63^\circ
\]

Right Triangle (Bottom):

We have a right triangle (one angle is \(90^\circ\)) with angles \((2x + 1)^\circ\), \((6x + 5)^\circ\), and \(90^\circ\).

Step 1: Recall the Triangle Sum Theorem

The sum of the angles in a triangle is \(180^\circ\), so \((2x + 1)^\circ + (6x + 5)^\circ + 90^\circ = 180^\circ\).

Step 2: Combine like terms

Combine the \(x\) terms and the constant terms:
\[
(2x + 6x) + (1 + 5) + 90^\circ = 180^\circ
\]
\[
8x + 6 + 90 = 180
\]
\[
8x + 96 = 180
\]

Step 3: Solve for \(x\)

Subtract 96 from both sides:
\[
8x = 180 - 96 = 84
\]
Then, divide both sides by 8:
\[
x = \frac{84}{8} = 10.5
\]

Now, we can find the angles:

• \((2x + 1)^\circ = 2(10.5) + 1 = 22^\circ\)
• \((6x + 5)^\circ = 6(10.5) + 5 = 68^\circ\)

Summary of Answers:

• Problem 5: \(x = 83^\circ\)
• Problem 6: \(x = 63^\circ\)
• Right Triangle: \(x = 10.5\), angles are \(22^\circ\), \(68^\circ\), and \(90^\circ\)