Question Video: Constructing Congruent Angles | Nagwa Question Video: Constructing Congruent Angles | Nagwa

# Question Video: Constructing Congruent Angles Mathematics • First Year of Preparatory School

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Which of these two figures shows the steps for constructing a congruent angle? [A] Figure I [B] Figure II

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### Video Transcript

Which of these two figures shows the steps for constructing a congruent angle? Figure I or figure II.

In this question, weβre given two figures and we need to determine which of the figures demonstrates the steps of duplicating an angle. We note that constructing means with a compass and a straight edge.

Letβs start by saying that these constructions are supposed to duplicate angle π·πΈπΉ. We can then recall that to duplicate this angle, we need to start by drawing a ray in the plane, say the ray from π΄ through πΊ, where π΄ will be the vertex of the angle we duplicate. We can see that both constructions have such a ray if we add a point πΊ to each figure as shown.

The next step in our construction is to trace a circle centered at πΈ that intersects the sides of the angle we want to duplicate at two points we will label π· prime and πΉ prime. After this, we need to trace a congruent circle centered at π΄. In the diagrams, we only sketch an arc of the circle to keep the construction clean. We call the point of intersection between the ray and the circle π΅. The final step in the construction is to trace a circle of radius π· prime πΉ prime centered at π΅. We can call the point of intersection of the two circles πΆ as shown.

We can conclude that the angles are congruent because triangle πΉ prime πΈπ· prime is congruent to triangle πΆπ΄π΅ by the side-side-side criterion. We see that this is only the case in the first figure.

For due diligence, we can check what the second construction gives us. We see that we have the arcs of two congruent circles centered at πΆ and π΅. This gives us the following pairs of lines of the same length, since they are radii of the congruent circles. If we connect π΄ to the point of intersection of these circles, π», as shown, then we can note that we have two congruent triangles. This means that all of the corresponding angles of the two triangles must be congruent. We can then note that this means that this gives us the angle bisector. Hence, the answer is that only figure I shows the steps for constructing a congruent angle.

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