Question

1. m =

n =
p =

1. d =

Explanation:

Step 1: Analyze the first figure (for m, n, p)

The figure appears to be a trapezoid with a midline (or a line segment parallel to the bases) and some congruent sides (marked with ticks) and angle markings. Also, there are two bases: 36 cm and 48 cm. The segment \( p \) is likely the midline of the trapezoid. The formula for the midline (or median) of a trapezoid is \( \text{Midline} = \frac{\text{Base}_1 + \text{Base}_2}{2} \). For angles, since the sides are marked as congruent (isosceles trapezoid? Wait, no, the sides with ticks: the non - parallel sides? Wait, actually, the line \( n \) and \( m \): looking at the angle markings, the triangle at the bottom: the angles are \( 73^\circ \) and \( 51^\circ \), but maybe the segment \( n \) and \( m \) are related to the sides. Wait, first, let's handle \( p \):

Base 1 = 36 cm, Base 2 = 48 cm. So \( p=\frac{36 + 48}{2}=\frac{84}{2}=42 \) cm.

For \( n \) and \( m \): The sides with ticks: the top side (length \( n \)) and the bottom side of the smaller trapezoid? Wait, actually, the figure has a triangle at the bottom? Wait, no, the left side is 48 cm, the right side of the trapezoid is 36 cm? Wait, no, the left side is 48 cm (marked with a single arrow), the right side of the trapezoid is 36 cm (marked with a single arrow). The line \( n \) is parallel to 36 cm? Wait, no, the top side of the smaller trapezoid (or the segment \( n \)): since the lines are marked with ticks (congruent segments), the segment \( n \) should be equal to 36 cm? Wait, no, maybe the triangle at the bottom: the two non - parallel sides of the trapezoid are marked with two ticks (congruent). Wait, maybe the segment \( m \) and \( n \): looking at the angle markings, the triangle has angles \( 73^\circ \) and \( 51^\circ \), but maybe the segment \( n \) is equal to 36 cm? Wait, no, let's re - examine.

Wait, the first figure: it's a trapezoid with a line parallel to the bases (the midline \( p \)). The top base is \( n \), the bottom base of the trapezoid is 48 cm, the top base of the big trapezoid is 36 cm? Wait, no, the left side is 48 cm (length), the right side of the trapezoid is 36 cm (length). The line \( n \) is parallel to 36 cm? Wait, maybe the segment \( n \) is equal to 36 cm? Wait, no, the ticks on the sides: the top side (with \( n \)) and the right side of the smaller trapezoid? Wait, maybe the triangle at the bottom: the two non - parallel sides of the trapezoid are congruent (marked with two ticks). The segment \( n \): since the top side of the trapezoid (36 cm) and the segment \( n \): wait, maybe the figure is a trapezoid with a midline, and the sides \( n \) and \( m \) are parts of the non - parallel sides. But maybe the angle \( 73^\circ \) and \( 51^\circ \): the sum of angles in a triangle is \( 180^\circ \), so the third angle is \( 180-(73 + 51)=56^\circ \)? No, maybe not. Wait, maybe the segment \( n \) is equal to 36 cm, and \( m \) is equal to... Wait, no, let's check the second figure.

Second figure: it's a trapezoid with a midline of length 24, one base is 13, find \( q \). The formula for the midline of a trapezoid is \( \text{Midline}=\frac{\text{Base}_1+\text{Base}_2}{2} \). Let \( \text{Base}_1 = 13 \), \( \text{Midline}=24 \), then \( 24=\frac{13 + q}{2} \). Multiply both sides by 2: \( 48=13 + q \), so \( q = 48 - 13=35 \)? Wait, no, wait: \( \text{Midline}=\frac{\text{Base}_1+\text{Base}_2}{2}\Rightarrow\text{Base}_2 = 2\times\text{Midline}-\text{Base}_1 \). So \( q=2\times24 - 13=48 - 13 = 35 \)? Wait, no, 2*24 is 48, 48 - 13 is 35? Wait, but let's go back to the first figure.

First figure:

- For \( p \): Midline of trapezoid, \( p=\frac{36 + 48}{2}=42 \) cm.

- For \( n \): The top base of the smaller trapezoid (or the segment parallel to 36 cm) should be equal to 36 cm? Wait, no, maybe the segment \( n \) is equal to 36 cm (since the sides are marked with ticks, congruent segments, so \( n = 36 \) cm.

- For \( m \): Similarly, the segment \( m \) should be equal to 36 cm? Wait, no, the left side is 48 cm, the right side is 36 cm? Wait, no, the left side is 48 cm (marked with a single arrow), the right side of the trapezoid is 36 cm (marked with a single arrow). The segment \( m \) and \( n \): since the lines are marked with two ticks (congruent), \( n = 36 \) cm, \( m = 36 \) cm? Wait, no, maybe the triangle at the bottom: the two non - parallel sides of the trapezoid are congruent, so the segments \( m \) and \( n \) are equal to 36 cm?

Now, the second figure:

We have a trapezoid with midline 24, one base 13, find \( q \). Using the midline formula: \( \text{Midline}=\frac{\text{Base}_1+\text{Base}_2}{2} \), so \( 24=\frac{13 + q}{2} \). Solving for \( q \):

Multiply both sides by 2: \( 48=13 + q \)

Subtract 13 from both sides: \( q = 48 - 13=35 \)

Wait, but let's confirm the first figure:

- \( m \): The angle - side relationship: the triangle at the bottom has angles \( 73^\circ \) and \( 51^\circ \), but maybe the segment \( m \) is equal to 36 cm (since the sides are congruent).

- \( n \): 36 cm.

- \( p \): 42 cm.

Second figure:

- \( q \): 35.

Wait, but let's do the first figure step - by - step:

Step 1: Calculate \( p \) (midline of trapezoid)

The formula for the midline (median) of a trapezoid is \( p=\frac{\text{length of base}_1+\text{length of base}_2}{2} \). Here, base\(_1 = 36\) cm and base\(_2 = 48\) cm.

\( p=\frac{36 + 48}{2}=\frac{84}{2}=42 \) cm.

Step 2: Determine \( n \)

The segment \( n \) is parallel to the base of length 36 cm and the sides adjacent to \( n \) are marked as congruent (ticks), so \( n = 36 \) cm.

Step 3: Determine \( m \)

Similarly, the segment \( m \) is parallel to the base of length 36 cm and the sides adjacent to \( m \) are marked as congruent (ticks), so \( m = 36 \) cm.

Step 4: Calculate \( q \) (second figure)

The formula for the midline of a trapezoid is \( \text{midline}=\frac{\text{base}_1+\text{base}_2}{2} \). Here, midline = 24, base\(_1 = 13\), and base\(_2 = q\).

Rearranging the formula: \( q = 2\times\text{midline}-\text{base}_1 \)

Substitute the values: \( q=2\times24 - 13=48 - 13 = 35 \)

Answer:

• For the first figure: \( m = 36\) cm, \( n = 36\) cm, \( p = 42\) cm.
• For the second figure: \( q = 35\) cm.