Question

1. identify two statements that contradict each other.

i. ∠m is an obtuse angle.
ii. m∠m + m∠p = 90
iii. 180 − m∠m = 25
iv. m∠p = 120
a i and ii c i and iv
b i and iii d iii and iv

1. which conclusion can you make from the given statements?

if a transversal crosses parallel lines, then alternate exterior angles are congruent. line t crosses parallel lines m and n.
image of transversal t crossing parallel lines m and n with angles labeled 1,2,3,4 on m and 5,6,7,8 on n
a ∠1 is congruent to ∠7
b ∠1 is congruent to ∠5
c ∠4 is congruent to ∠8
d ∠4 is congruent to ∠6

1. what is the value of x?

image of a straight line with a right angle and angles (5y + 10)° and (3x + 7)°
a 16
b 20
c 21
d 28

1. complete the proof using the statements and reasons below.

given: ∠1 and ∠2 are vertical angles
m∠2 + m∠3 = 90
prove: m∠1 + m∠3 = 90
statements | reasons

1. ∠1 and ∠2 are vertical angles; m∠2 + m∠3 = 90 | 1) given
2. | 2) vertical angles are congruent
3. | 3)

box with options: substitution; m∠1 = m∠3; vertical angles are congruent; m∠1 = m∠2; m∠1 + m∠3 = 90

1. fill in the blanks to write the contrapositive of the following statement and complete its proof.

if a polygon has more than three sides, then it is not a triangle.
boxes with options: true, false, is not, conditional statement, has more than, is, converse, has exactly
contrapositive: if a polygon blank a triangle, then the polygon blank three sides.
since a triangle has exactly three sides, the contrapositive is blank.
since the contrapositive is blank, the blank must be true

Explanation:

Question 36

Step 1: Analyze Statement I

An obtuse angle is greater than \(90^\circ\) and less than \(180^\circ\). So, if \(\angle M\) is obtuse, \(m\angle M>90^\circ\).

Step 2: Analyze Statement II

\(m\angle M + m\angle P=90^\circ\) implies \(m\angle M=90^\circ - m\angle P\), so \(m\angle M<90^\circ\) (since \(m\angle P>0^\circ\)). This contradicts Statement I (obtuse angle > \(90^\circ\)).

Step 3: Analyze Statement III

From \(180 - m\angle M = 25\), we solve for \(m\angle M\): \(m\angle M=180 - 25 = 155^\circ\), which is obtuse (consistent with Statement I).

Step 4: Analyze Statement IV

\(m\angle P = 120^\circ\), and from Statement II \(m\angle M + 120^\circ=90^\circ\) would imply \(m\angle M=- 30^\circ\) (impossible), but Statement I is about \(\angle M\) being obtuse, not directly about \(m\angle P\). The contradiction between I and II is clear.

Step 1: Recall the Theorem

The theorem states: If a transversal crosses parallel lines, then alternate exterior angles are congruent.

Step 2: Identify Angles

For lines \(m\) and \(n\) (parallel) cut by transversal \(t\), alternate exterior angles: \(\angle 4\) and \(\angle 8\) are not alternate exterior. \(\angle 1\) and \(\angle 5\) are corresponding (not alternate exterior). \(\angle 1\) and \(\angle 7\): \(\angle 1\) and \(\angle 3\) are vertical, \(\angle 3\) and \(\angle 7\) are corresponding? Wait, no. Wait, alternate exterior angles: \(\angle 4\) and \(\angle 6\) are alternate interior? Wait, no. Wait, the correct pair: \(\angle 4\) and \(\angle 6\) are alternate interior angles? Wait, no, let's re - examine. The parallel lines are \(m\) and \(n\), transversal \(t\). Alternate exterior angles: \(\angle 1\) (exterior to \(m\) and \(n\) on top) and \(\angle 8\) (exterior on bottom)? No, wait the correct application: \(\angle 4\) and \(\angle 6\): \(\angle 4\) is on line \(m\), \(\angle 6\) on line \(n\), on opposite sides of transversal \(t\), and inside the two lines? No, \(\angle 4\) and \(\angle 6\) are alternate interior angles? Wait, no, the theorem says alternate exterior angles. Wait, \(\angle 1\) and \(\angle 5\) are corresponding. Wait, the correct answer: \(\angle 4\) and \(\angle 6\) are alternate interior? No, the option D: \(\angle 4\) is congruent to \(\angle 6\) (alternate interior angles, which are congruent when lines are parallel). Wait, the theorem given is about alternate exterior, but maybe a mis - label? Wait, no, let's check the angles: \(\angle 4\) and \(\angle 6\): line \(m\) and \(n\) are parallel, transversal \(t\). \(\angle 4\) and \(\angle 6\) are alternate interior angles (between \(m\) and \(n\), on opposite sides of \(t\)), so they are congruent. \(\angle 1\) and \(\angle 5\) are corresponding (congruent), but option B is \(\angle 1\cong\angle 5\), but wait the theorem says alternate exterior. Wait, maybe the diagram: \(\angle 4\) and \(\angle 6\): \(\angle 4\) is below \(m\), left of \(t\); \(\angle 6\) is below \(n\), right of \(t\)? No, maybe I made a mistake. Wait, the correct answer is D: \(\angle 4\) is congruent to \(\angle 6\) (alternate interior angles, which are congruent for parallel lines).

Step 1: Identify Angle Relationships

We have a right angle (the box), so the sum of \((5y + 10)^\circ\), the right angle (\(90^\circ\)), and \((3x + 7)^\circ\) is \(180^\circ\) (straight line). Also, \((5y + 10)^\circ\) and \((3x + 7)^\circ\) are vertical angles? Wait, no, the angle \(y^\circ\) and the right angle and \((3x + 7)^\circ\): Wait, the angle \(y^\circ\) is adjacent to the right angle, so \(y + 90+(3x + 7)=180\)? No, wait, the vertical angle to \((5y + 10)^\circ\) is \((3x + 7)^\circ\), so \(5y+10 = 3x + 7\). Also, \(y + 90+(3x + 7)=180\) (since they form a straight line). Wait, let's solve:

From the straight line: \(y+90+(3x + 7)=180\Rightarrow y + 3x=83\).

From vertical angles: \(5y + 10=3x + 7\Rightarrow 5y-3x=- 3\).

Now we have a system:
\(

$$\begin{cases}y + 3x=83\\5y-3x=-3\end{cases}$$

\)

Add the two equations: \(6y = 80\)? No, that can't be. Wait, maybe the angle \(y^\circ\) and \((5y + 10)^\circ\) are supplementary to the right angle? Wait, the right angle is \(90^\circ\), so \(y+(5y + 10)=90\) (since they are adjacent to the right angle and form a straight line? Wait, no, the diagram: the angle \(y^\circ\), the right angle, and \((3x + 7)^\circ\) are on a straight line, so \(y + 90+(3x + 7)=180\Rightarrow y+3x = 83\). Also, \((5y + 10)^\circ\) and \((3x + 7)^\circ\) are vertical angles, so \(5y + 10=3x + 7\Rightarrow 5y-3x=-3\).

Now solve the system:
From the first equation: \(y = 83 - 3x\).

Substitute into the second equation: \(5(83 - 3x)-3x=-3\Rightarrow415-15x - 3x=-3\Rightarrow415-18x=-3\Rightarrow-18x=-418\)? No, that's wrong. Wait, maybe the angle \(y^\circ\) and \((5y + 10)^\circ\) are supplementary (since they are adjacent and form a straight line with the right angle? Wait, no, the vertical angle of \((5y + 10)^\circ\) is \((3x + 7)^\circ\), so \(5y + 10=3x + 7\). Also, \(y + (3x + 7)=90\) (since \(y\) and \((3x + 7)\) are adjacent to the right angle, forming a right angle? Wait, the right angle is \(90^\circ\), so \(y+(3x + 7)=90\Rightarrow y + 3x=83\). And \(5y + 10=3x + 7\Rightarrow 5y-3x=-3\).

Now, from \(y = 83 - 3x\), substitute into \(5y-3x=-3\):

\(5(83 - 3x)-3x=-3\)

\(415-15x-3x=-3\)

\(415 - 18x=-3\)

\(-18x=-418\)

\(x=\frac{418}{18}\approx23.22\) (not matching options). Wait, maybe I misread the diagram. The angle \(y^\circ\) and \((5y + 10)^\circ\) are supplementary (since they are adjacent and form a straight line), so \(y+(5y + 10)=180\Rightarrow6y + 10=180\Rightarrow6y = 170\Rightarrow y=\frac{85}{3}\approx28.33\). Then, the angle \((3x + 7)^\circ\) is equal to \(y^\circ\) (since they are adjacent to the right angle? No, the right angle is \(90^\circ\), so \(y + (3x + 7)+90 = 180\Rightarrow y+3x=83\). If \(y=\frac{85}{3}\), then \(3x=83-\frac{85}{3}=\frac{249 - 85}{3}=\frac{164}{3}\Rightarrow x=\frac{164}{9}\approx18.22\) (still not matching). Wait, the options are 16, 20, 21, 28. Maybe the angle \((5y + 10)^\circ\) and \((3x + 7)^\circ\) are equal (vertical angles), and \(y + (3x + 7)=90\) (since \(y\) and \((3x + 7)\) are complementary to the right angle). So we have:

1. \(5y + 10=3x + 7\)
2. \(y+(3x + 7)=90\)

From equation 2: \(y + 3x=83\Rightarrow3x=83 - y\). Substitute into equation 1:

\(5y + 10=83 - y+7\)

\(5y + 10=90 - y\)

\(6y=80\) (no, 90 - 10 = 80? Wait, 83+7 = 90, so \(5y + 10=90 - y\Rightarrow6y=80\Rightarrow y=\frac{40}{3}\approx13.33\). Then \(3x=83-\frac{40}{3}=\frac{249 - 40}{3}=\frac{209}{3}\approx69.67\Rightarrow x\approx23.22\). This is wrong. Maybe the right angle is between \(y\) and \((3x + 7)\), so \(y + (3x + 7)=90\), and \((5y + 10)\) is vertical to \((3…

Answer:

A. I and II

Question 37