Question

what is the value of p?
p =
submit

Explanation:

Step1: Identify angle relationship

The two right triangles are congruent (HL, since hypotenuses are equal and legs are right angles), so the angles \( p + 66^\circ \) and \( 7p \) are equal? Wait, no—wait, the angles at \( T \): Wait, actually, the sum of angles in a straight line? Wait, no, the two right angles and the angles \( p + 66^\circ \) and \( 7p \) should add up? Wait, no, looking at the diagram, the two right angles (at \( U \) and \( S \)) and the angles \( p + 66^\circ \) and \( 7p \) form a linear pair? Wait, no, actually, the two triangles are congruent (legs are right angles, hypotenuses are equal as marked with the same tick), so the angles \( p + 66^\circ \) and \( 7p \) are supplementary? Wait, no, wait, the straight line at \( T \): the two right angles (90 each) and \( p + 66 \) and \( 7p \) should add up to \( 360^\circ \)? No, wait, no—wait, the angles at \( T \): the two right angles (90° each) and the angles \( p + 66^\circ \) and \( 7p \) are around point \( T \), but actually, the two triangles are congruent, so the angles \( p + 66^\circ \) and \( 7p \) are supplementary? Wait, no, let's re-examine.

Wait, the two triangles \( RTU \) and \( RTS \) are right triangles with \( RU = RS \) (marked with ticks) and \( RT \) common, so by HL congruence, they are congruent. Therefore, the angles \( \angle RTU \) and \( \angle RTS \) are equal? Wait, no, \( \angle RTU = p + 66^\circ \) and \( \angle RTS = 7p \). Wait, but since the two right angles are at \( U \) and \( S \), and \( RU = RS \), \( RT \) is common, so triangles \( RTU \cong RTS \) (HL). Therefore, \( \angle RTU = \angle RTS \)? No, wait, no—wait, the angles at \( T \): \( \angle UT S \) is a straight line? Wait, no, \( U \) and \( S \) are on a straight line? Wait, the arrows at \( U \) and \( S \) suggest a straight line, so \( \angle UTS \) is a straight angle (180°)? No, the two right angles (90° each) and \( p + 66^\circ \) and \( 7p \) are adjacent angles forming a full angle? No, that can't be. Wait, maybe the sum of \( p + 66^\circ \) and \( 7p \) is 180°? Wait, no, let's think again.

Wait, the two right angles (90° each) and the angles \( p + 66^\circ \) and \( 7p \) are around point \( T \), but actually, the two triangles are congruent, so \( p + 66^\circ + 7p = 180^\circ \)? Wait, no, that doesn't make sense. Wait, maybe the angles \( p + 66^\circ \) and \( 7p \) are supplementary? Wait, let's set up the equation. Wait, the sum of angles on a straight line: if \( U \) and \( S \) are on a straight line, then the angles at \( T \): \( 90^\circ + (p + 66^\circ) + 90^\circ + 7p = 360^\circ \)? No, that would be around a point, but \( U \) and \( S \) are on a straight line, so the angles on one side of the line: \( 90^\circ + (p + 66^\circ) + 7p + 90^\circ = 360^\circ \)? No, that's a full circle. Wait, no, the diagram shows \( U \) and \( S \) on a straight line (opposite arrows), so \( \angle UTS \) is 180°, so the angles at \( T \) on that line: \( 90^\circ + (p + 66^\circ) + 7p + 90^\circ = 360^\circ \)? No, that's not right. Wait, maybe the two angles \( p + 66^\circ \) and \( 7p \) are equal? Wait, no, if the triangles are congruent, then \( \angle RTU = \angle RTS \), but \( \angle RTU = p + 66^\circ \) and \( \angle RTS = 7p \), so \( p + 66 = 7p \)? Wait, solving that: \( 66 = 6p \), \( p = 11 \). Wait, but let's check: \( p + 66 = 77 \), \( 7p = 77 \), so 77 + 77 + 90 + 90 = 334, which is not 360. Wait, that's wrong.

Wait, maybe the angles \( p + 66^\circ \) and \( 7p \) are supplementary (sum to 180°) because they are on a straight line with the two right angles? Wait, no, the two right angles are 90 each, so 90 + 90 = 180, so the remaining two angles \( p + 66 \) and \( 7p \) must also sum to 180° (since total around point T is 360°, 360 - 180 = 180). So \( (p + 66) + 7p = 180 \).

Step2: Solve the equation

Combine like terms: \( 8p + 66 = 180 \)

Subtract 66: \( 8p = 180 - 66 = 114 \)? Wait, no, 180 - 66 is 114? Wait, 180 - 60 is 120, minus 6 is 114. Then \( p = 114 / 8 = 14.25 \)? No, that can't be. Wait, maybe my initial assumption is wrong.

Wait, another approach: the two triangles are right triangles with \( RU \perp UT \) and \( RS \perp ST \), so \( RU \) and \( RS \) are both perpendicular to \( US \), so \( RU \parallel RS \)? No, they are the same line? Wait, no, \( U \) and \( S \) are on a straight line, so \( US \) is a straight line, and \( RU \) and \( RS \) are both perpendicular to \( US \), so \( RU \) and \( RS \) are parallel? No, they are both perpendicular to \( US \), so they are colinear? Wait, the diagram shows \( RU \) and \( RS \) as two lines from \( R \) to \( U \) and \( S \), with \( U \) and \( S \) on a straight line, and both \( RU \) and \( RS \) perpendicular to \( US \), so \( RU \) and \( RS \) are the same line? But the diagram shows them as two different lines with a common vertex \( R \). Wait, the ticks on \( RU \) and \( RS \) mean they are equal in length, not that they are the same line. So \( RU = RS \), and both are perpendicular to \( US \), so \( \triangle RTU \) and \( \triangle RTS \) are right triangles with \( RU = RS \) and \( RT \) common, so HL congruence, so \( \angle RTU = \angle RTS \). Wait, \( \angle RTU = p + 66^\circ \) and \( \angle RTS = 7p \), so \( p + 66 = 7p \), so \( 66 = 6p \), \( p = 11 \). Then \( \angle RTU = 77^\circ \), \( \angle RTS = 77^\circ \), and the two right angles (90 each) sum to 180, so total around T is 77 + 77 + 90 + 90 = 334, which is not 360. That's a problem.

Wait, maybe the angle \( p + 66^\circ \) and \( 7p \) are vertical angles? No. Wait, maybe the straight line at \( R \): no, \( R \) is above. Wait, maybe the sum of \( p + 66 \) and \( 7p \) is 180 because they are supplementary. Let's try that: \( p + 66 + 7p = 180 \), \( 8p = 114 \), \( p = 14.25 \). No, that's not an integer. Wait, maybe the diagram has a typo, but assuming the correct relationship is \( p + 66 + 7p = 180 \) (since the two right angles are 90 each, so 90 + 90 = 180, and the other two angles sum to 180 for a full circle? No, 360 - 180 = 180). Wait, maybe I made a mistake.

Wait, let's re-express the diagram: at point \( T \), we have four angles: two right angles (90° each) and two angles \( p + 66^\circ \) and \( 7p \). The sum of angles around a point is 360°, so:

\( 90 + (p + 66) + 7p + 90 = 360 \)

Simplify: \( 90 + 90 + p + 66 + 7p = 360 \)

\( 180 + 66 + 8p = 360 \)

\( 246 + 8p = 360 \)

\( 8p = 360 - 246 = 114 \)

\( p = 114 / 8 = 14.25 \). No, that's not nice. Wait, maybe the angles \( p + 66 \) and \( 7p \) are equal, so \( p + 66 = 7p \), \( 6p = 66 \), \( p = 11 \). Then the total angles: 90 + 77 + 77 + 90 = 334, which is 26 less than 360. That can't be. Maybe the diagram is different. Wait, maybe the angle \( p + 66^\circ \) and \( 7p \) are complementary? No. Wait, maybe the two right angles are not both 90? No, the red squares mean right angles (90°).

Wait, maybe the correct equation is \( p + 66 + 7p = 180 \) (since the two right angles are on a straight line, so the other two angles are supplementary). Let's solve that:

\( 8p + 66 = 180 \)

\( 8p = 114 \)

\( p = 14.25 \). But that's a decimal. Maybe the problem has a typo, but assuming the intended relationship is \( p + 66 = 7p \), then \( p = 11 \). Let's check with \( p = 11 \): \( p + 66 = 77 \), \( 7p = 77 \), so they are equal, which would make sense if the triangles are congruent, so the angles at \( T \) are equal. Then the total around \( T \) is 90 + 77 + 77 + 90 = 334, which is incorrect, but maybe the diagram is not to scale. Alternatively, maybe the angle \( p + 66^\circ \) and \( 7p \) are vertical angles, but no. Wait, maybe the sum is 90? No. Wait, maybe the angle \( 7p \) is equal to \( p + 66 \), so \( 7p = p + 66 \), \( 6p = 66 \), \( p = 11 \). That must be it, because the triangles are congruent, so the angles opposite equal sides are equal. So the answer is 11.

Step1: Set angles equal (congruent triangles)

Since \( \triangle RTU \cong \triangle RTS \) (HL: right angles, equal hypotenuses \( RU = RS \), common leg \( RT \)), their corresponding angles \( \angle RTU \) and \( \angle RTS \) are equal. Thus, \( p + 66 = 7p \).

Step2: Solve for \( p \)

Subtract \( p \) from both sides: \( 66 = 6p \).
Divide by 6: \( p = \frac{66}{6} = 11 \).

Answer:

\( \boxed{11} \)