Question

geometry 3.1 - 3.3 quiz
questions #10 - #15 refer to the figure on the right.
for questions #10 - #13, determine which lines, if any are parallel. state the postulate or theorem that justifies your answer.

1. $\angle 1\cong\angle 6$ 11. $\angle 2\cong\angle 3$
2. $\angle 4\cong\angle 9$ 13. $m\angle 7 + m\angle 8 = 180$
3. given $g\parallel h$ and $m\angle 8 = 64$, find $m\angle 5$.
4. if $m\angle 2 = 5x - 17$ and $m\angle 7 = 3x + 35$, find the value of $x$ so that $g\parallel h$.
5. the measures of the three angles of a triangle are $x + 10$, $x - 20$, and $x + 25$. find $x$.
6. find $x$ and $y$.

Explanation:

Step1: Use the property of parallel lines and transversals

When two parallel lines are cut by a transversal, corresponding angles are congruent, alternate interior angles are congruent, and consecutive interior angles are supplementary.

Step2: Analyze each question

• Question 10:

Since \(\angle1\cong\angle6\), by the Alternate Exterior Angles Theorem, the lines are parallel.

• Question 11:

Since \(\angle2\cong\angle3\), by the Alternate Interior Angles Theorem, the lines are parallel.

• Question 12:

Since \(\angle4\cong\angle9\), by the Corresponding Angles Postulate, the lines are parallel.

• Question 13:

Since \(m\angle7 + m\angle8=180\), by the Consecutive Interior Angles Theorem, the lines are parallel.

• Question 14:

Given \(g\parallel h\) and \(m\angle8 = 64\). \(\angle8\) and \(\angle5\) are supplementary (consecutive interior angles). So \(m\angle5=180 - 64=116\).

• Question 15:

If \(g\parallel h\), then \(\angle2\cong\angle7\) (alternate interior angles). So \(5x-17 = 3x + 35\).
Subtract \(3x\) from both sides: \(2x-17=35\).
Add \(17\) to both sides: \(2x=52\).
Divide by \(2\): \(x = 26\).

• Question 16:

The sum of the angles in a triangle is \(180\). So \((x + 10)+(x - 20)+(x + 25)=180\).
Combine like terms: \(3x + 15=180\).
Subtract \(15\) from both sides: \(3x=165\).
Divide by \(3\): \(x = 55\).

• Question 17:

Assuming the triangle has an exterior angle \(x + 16\). By the Exterior Angle Theorem, \(x+16=x + y\). Also, if we assume other properties (since the figure is not clear), but if it's a simple case of an isosceles triangle (assuming \(y = 67\) and \(x+16=x + 67\) is wrong, but if we assume the triangle has angles \(x\), \(y\), and \(67\) and the exterior angle \(x + 16\). Then \(x+16=x + y\) (exterior angle theorem) gives \(y = 16\). And if it's a triangle with sum \(x + y+67 = 180\), substituting \(y = 16\) gives \(x=97\).

Answer:

1. Lines are parallel by Alternate Exterior Angles Theorem.
2. Lines are parallel by Alternate Interior Angles Theorem.
3. Lines are parallel by Corresponding Angles Postulate.
4. Lines are parallel by Consecutive Interior Angles Theorem.
5. \(m\angle5 = 116\)
6. \(x = 26\)
7. \(x = 55\)
8. \(x = 97,y = 16\) (assuming standard triangle - exterior angle relations as described in steps)