Question
Congruency of Triangles Practice
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Triangles upper M upper N upper O and upper P upper Q upper R are graphed side by side on a grid.
ΔOMN≅ΔRPQ
by the HL Theorem. If m∠O=57
degrees, then what is the measure of ∠P
?
(1 point)
degrees
Answers
Answer
that answers is wrong
Answered by
GPT-4o mini
In congruent triangles, corresponding angles are equal. Since ΔOMN is congruent to ΔRPQ (ΔOMN ≅ ΔRPQ), it follows that m∠O = m∠P.
Given that m∠O = 57 degrees, we can conclude that:
m∠P = 57 degrees.
Therefore, the measure of ∠P is 57 degrees.
Given that m∠O = 57 degrees, we can conclude that:
m∠P = 57 degrees.
Therefore, the measure of ∠P is 57 degrees.
Answered by
GPT-4o mini
I apologize for any confusion. Let's analyze the problem again.
If ΔOMN is congruent to ΔRPQ by the HL Theorem (Hypotenuse-Leg Theorem), it means that both triangles are right triangles and share congruent corresponding sides and angles.
Given that \( m∠O = 57^\circ \), we need to determine the measure of \( ∠P \).
In the case of right triangles:
1. Let's assume that \( ∠O \) is one of the non-right angles in triangle OMN.
2. In triangle RPQ, if we denote the right angle as \( ∠Q \) (assuming \( P \) and \( Q \) are the other two vertices), then the corresponding angle \( ∠P \) should be the other non-right angle.
For the angles in a triangle, the sum of the angles in a triangle equals 180 degrees. If we denote the right angle as 90 degrees, then:
\[ m∠M + m∠O + m∠N = 180^\circ \]
Since \( ∠N \) (the right angle) is 90 degrees, we can find the measure of the other angle:
\[ m∠M = 180^\circ - m∠O - 90^\circ = 180^\circ - 57^\circ - 90^\circ = 33^\circ \]
Since the two triangles are congruent, \( m∠M \) in triangle OMN corresponds to \( m∠R \) in triangle RPQ.
Thus, \( m∠P \) and \( m∠R \) together must equal 90 degrees, since \( ∠Q \) is a right angle (as we considered earlier).
If \( m∠R = 33^\circ \), then we follow that:
\[ m∠P = 90^\circ - m∠R = 90^\circ - 33^\circ = 57^\circ \]
But since both angles are actually \( 90^\circ - m∠O \):
So \( m∠P = 90^\circ - m∠O = 90^\circ - 57^\circ = 33^\circ \).
Thus, the measure of \( ∠P \) should be:
\[ m∠P = 33^\circ. \]
Thank you for your patience! So, the corrected answer should be **33 degrees**.
If ΔOMN is congruent to ΔRPQ by the HL Theorem (Hypotenuse-Leg Theorem), it means that both triangles are right triangles and share congruent corresponding sides and angles.
Given that \( m∠O = 57^\circ \), we need to determine the measure of \( ∠P \).
In the case of right triangles:
1. Let's assume that \( ∠O \) is one of the non-right angles in triangle OMN.
2. In triangle RPQ, if we denote the right angle as \( ∠Q \) (assuming \( P \) and \( Q \) are the other two vertices), then the corresponding angle \( ∠P \) should be the other non-right angle.
For the angles in a triangle, the sum of the angles in a triangle equals 180 degrees. If we denote the right angle as 90 degrees, then:
\[ m∠M + m∠O + m∠N = 180^\circ \]
Since \( ∠N \) (the right angle) is 90 degrees, we can find the measure of the other angle:
\[ m∠M = 180^\circ - m∠O - 90^\circ = 180^\circ - 57^\circ - 90^\circ = 33^\circ \]
Since the two triangles are congruent, \( m∠M \) in triangle OMN corresponds to \( m∠R \) in triangle RPQ.
Thus, \( m∠P \) and \( m∠R \) together must equal 90 degrees, since \( ∠Q \) is a right angle (as we considered earlier).
If \( m∠R = 33^\circ \), then we follow that:
\[ m∠P = 90^\circ - m∠R = 90^\circ - 33^\circ = 57^\circ \]
But since both angles are actually \( 90^\circ - m∠O \):
So \( m∠P = 90^\circ - m∠O = 90^\circ - 57^\circ = 33^\circ \).
Thus, the measure of \( ∠P \) should be:
\[ m∠P = 33^\circ. \]
Thank you for your patience! So, the corrected answer should be **33 degrees**.
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