Question

congruence and similarity pretest
geometry cr highland / module 3: congruence and similarity

1. what is the measure of angle khj?

(figure of a triangle jhi with angle at j being 85°, angle at i being 47°, and point k on the extension of hi beyond h)

Explanation:

Step1: Find angle at H in triangle JHI

In triangle \( JHI \), the sum of angles in a triangle is \( 180^\circ \). So angle \( \angle JHI = 180^\circ - 85^\circ - 47^\circ \)
\( \angle JHI = 180 - 85 - 47 = 48^\circ \)

Step2: Find angle \( \angle KHJ \)

Since \( K, H, I \) are colinear, \( \angle KHJ + \angle JHI = 180^\circ \) (linear pair). So \( \angle KHJ = 180^\circ - \angle JHI \)
\( \angle KHJ = 180 - 48 = 132^\circ \) Wait, no, wait. Wait, maybe I made a mistake. Wait, actually, \( \angle KHJ \) and \( \angle JHI \) are supplementary? Wait, no, looking at the diagram, \( K - H - I \) is a straight line. So \( \angle KHJ + \angle JHI = 180^\circ \)? Wait, no, angle at H: \( \angle KHJ \) and \( \angle JHI \) are adjacent and form a linear pair? Wait, no, maybe I misread the triangle. Wait, triangle JHI: angles at J is 85, at I is 47, so angle at H (JHI) is 180 - 85 -47 = 48. Then, since K-H-I is a straight line, angle KHJ is 180 - angle JHI? Wait, no, angle KHJ and angle JHI: if H is on the line KI, then angle KHJ + angle JHI + angle... Wait, no, the diagram: K---H---I is a straight line. J is above H. So triangle JHI has vertices J, H, I. So angle at H in triangle is \( \angle JHI \), and angle \( \angle KHJ \) is adjacent to \( \angle JHI \) forming a linear pair? Wait, no, \( \angle KHJ \) and \( \angle JHI \): if K-H-I is straight, then \( \angle KHJ + \angle JHI = 180^\circ \)? Wait, no, \( \angle KHJ \) is at H between K and J, and \( \angle JHI \) is at H between J and I. So yes, they are supplementary. Wait, but earlier calculation: angle JHI is 48, so angle KHJ is 180 - 48 = 132? Wait, but that seems wrong. Wait, maybe I messed up the triangle angles. Wait, sum of angles in triangle: 85 + 47 + angle JHI = 180. So 85 +47=132, so angle JHI=48. Then, since K-H-I is straight, angle KHJ + angle JHI = 180? Wait, no, angle KHJ and angle JHI: if you look at the diagram, K is to the left of H, I to the right. J is above H. So angle KHJ is the angle between KH and JH, and angle JHI is between JH and HI. So they are adjacent and form a linear pair, so their sum is 180. So angle KHJ = 180 - angle JHI = 180 - 48 = 132? Wait, but that seems big. Wait, maybe the question is different. Wait, maybe the triangle is isoceles? No, the problem is to find angle KHJ. Wait, maybe I made a mistake in the triangle angle sum. Wait, 85 + 47 = 132, 180 - 132 = 48. So angle JHI is 48. Then angle KHJ is 180 - 48 = 132? Wait, but maybe the diagram is different. Wait, maybe the angle at I is 47, angle at J is 85, so angle at H (JHI) is 48. Then, since K-H-I is a straight line, angle KHJ is 180 - 48 = 132? Wait, but maybe the answer is 48? No, that can't be. Wait, no, maybe I got the angle wrong. Wait, maybe angle KHJ is equal to angle at I? No, that doesn't make sense. Wait, maybe the triangle is similar or congruent? Wait, no, the question is about angle KHJ. Wait, let's re-express:

In triangle \( JHI \):

\( \angle J = 85^\circ \), \( \angle I = 47^\circ \)

Sum of angles in a triangle: \( \angle J + \angle I + \angle JHI = 180^\circ \)

So \( \angle JHI = 180 - 85 - 47 = 48^\circ \)

Now, \( K, H, I \) are collinear, so \( \angle KHJ + \angle JHI = 180^\circ \) (linear pair)

Wait, no, \( \angle KHJ \) and \( \angle JHI \) are adjacent angles forming a linear pair, so their sum is \( 180^\circ \)

Thus, \( \angle KHJ = 180^\circ - \angle JHI = 180 - 48 = 132^\circ \)? Wait, but that seems incorrect. Wait, maybe the diagram is such that \( \angle KHJ \) is equal to \( \angle J \) or something? No, that's not a theorem. Wait, maybe I misread the angle at I. Wait, the diagram shows angle at I is 47, angle at J is 85. So angle at H is 48. Then, since K-H-I is straight, angle KHJ is 180 - 48 = 132. But that seems large. Wait, maybe the question is asking for angle KHJ, which is actually equal to 180 - (85 + 47)? Wait, no, 85 +47=132, 180-132=48. Wait, no, that's the angle at H. Wait, I'm confused. Wait, maybe the diagram is a triangle where KHJ is an external angle? Wait, no. Wait, let's look again. The problem is to find angle KHJ. The triangle JHI has angles 85 (at J), 47 (at I), so angle at H (JHI) is 48. Then, since K, H, I are on a straight line, angle KHJ is supplementary to angle JHI? Wait, no, angle KHJ and angle JHI: if you have a straight line KI, and J above H, then angle KHJ is the angle between KH and JH, and angle JHI is between JH and HI. So they are adjacent and form a linear pair, so sum to 180. So angle KHJ = 180 - 48 = 132. But that seems odd. Wait, maybe the answer is 48? No, that can't be. Wait, maybe I made a mistake in the triangle angle sum. Wait, 85 + 47 = 132, 180 - 132 = 48. So angle JHI is 48. Then angle KHJ is 180 - 48 = 132. So the measure of angle KHJ is 132 degrees? Wait, but maybe the diagram is different. Wait, maybe the angle at I is 47, angle at J is 85, so angle at H is 48, and angle KHJ is equal to angle at J + angle at I? Wait, that's the exterior angle theorem! Oh! Right! The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. So angle KHJ is an exterior angle to triangle JHI at vertex H. So exterior angle at H: angle KHJ = angle at J + angle at I = 85 + 47 = 132. Yes! That's the exterior angle theorem. So that's the correct approach. So step 1: recall exterior angle theorem. Step 2: apply it. So angle KHJ is equal to the sum of the two remote interior angles, which are angle at J (85) and angle at I (47). So 85 + 47 = 132.

Step1: Apply Exterior Angle Theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For triangle \( JHI \), \( \angle KHJ \) is an exterior angle at vertex \( H \), and the two non - adjacent interior angles are \( \angle J = 85^\circ \) and \( \angle I = 47^\circ \).

Step2: Calculate the measure of \( \angle KHJ \)

Using the exterior angle theorem, we have \( \angle KHJ=\angle J+\angle I \)
\( \angle KHJ = 85^\circ+ 47^\circ=132^\circ \)

Answer:

\( 132^\circ \)