Question

Question was provided via image upload.

Explanation:

Identify the given geometric values

Using the Triangle Angle Sum Theorem and Supplementary Angles knowledge points
Let the points on the straight horizontal line from left to right be \(A\), \(P\), and \(D\).
The given values from the diagram are:

• \(\angle B = 30^\circ\)
• \(\angle BPC = 80^\circ\)
• \(\angle CPD = 50^\circ\)
• \(\angle C = 30^\circ\)
• Points \(A\), \(P\), and \(D\) lie on a straight line, meaning \(\angle APD = 180^\circ\).

Calculate the missing angles around point P

Using the Supplementary Angles knowledge point

$$
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$$

Calculate the remaining interior angles of the triangles

Using the Triangle Angle Sum Theorem knowledge point
For \(\triangle ABP\):

$$
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$$

For \(\triangle CDP\):

$$
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$$

Determine the relationship between the two triangles

Using the Triangle Congruence knowledge point

• \(\angle A = \angle D = 100^\circ\)
• \(\angle B = \angle C = 30^\circ\)
• \(\angle APB = \angle DPC = 50^\circ\)

The triangles are similar by Angle-Angle (AA) similarity:

$$ \triangle ABP \sim \triangle DCP $$

Answer:

Based on the geometric relationships shown in the diagram:

1. Angles on the straight line at the bottom vertex \(P\):

$$ \angle APB = 180^\circ - (80^\circ + 50^\circ) = 50^\circ $$

1. Interior angles of the left triangle \(\triangle ABP\):

$$ \angle A = 180^\circ - (30^\circ + 50^\circ) = 100^\circ $$

1. Interior angles of the right triangle \(\triangle CDP\):

$$ \angle D = 180^\circ - (30^\circ + 50^\circ) = 100^\circ $$

1. Triangle Similarity:

Since all corresponding angles are equal (\(100^\circ\), \(30^\circ\), and \(50^\circ\)), the two triangles are similar by Angle-Angle (AA) similarity:

$$ \triangle ABP \sim \triangle DCP $$