Problem 5
ABCDE and CPQRS are regular pentagons. If AR and EQ intersect at T, find ∠RTE.
Here’s a simulation that demonstrates the problem.
You can drag the the point P and observe ∠RTE. Does its measure remain the same? What is the measure?
Solution
As usual, I won’t provide the full solution here, but only give you some cues that’ll help you solve the problem yourself.
You’ll need to know two things here.
Congruent Triangles
If the corresponding sides and angles of two triangles are equal, then they are said to be congruent.
And, if we’re able to prove that two triangles are congruent, then their corresponding parts (sides and angles) are equal.
There are various ways to prove the congruence of two triangles, one of them being the Side-Angle-Side or SAS rule.
That is, if for two triangles ABC and DEF that AB = DE, BC = EF, and ∠B = ∠E, then the two triangles would be congruent.
And, this would mean that AC = DF, ∠A = ∠D, and ∠C = ∠F (i.e., the corresponding parts are equal).
You can observe this in the following simulation.
Drag E and F such that AB = DE and AC = DF. Does BC equal EF in that case?
Concyclic Points
Four points A, B, C, and D are said to be concyclic if a circle can pass through them.
A circle can always pass through three non-collinear points (say A, B, and C). But the fourth one (D) to also lie on the same circle, it must satisfy the following conditions:
∠ADB = ∠ACB (assuming C and D lie on the same side of AB)
Conversely, if A, B, C, and D are four points such that ∠ADB = ∠ACB (where C and D are on the same side of AB), then these points are concyclic.
You can observe this in the simulation below.
Try dragging D to a position where ∠ADB = ∠ACB. Does it end up lying on the circle? Also, would ∠DAB and ∠DBC be also equal in that case?
Now, let’s come back to the problem. To proceed, you have to find two triangles that are congruent. (You’ll have to construct them.)
Next, the corresponding angles of these triangles would be equal, which would lead you to prove that four particular points in the figure are concyclic.
Finally, this would lead you to prove that two particular angles are equal, one of which would be known to you, and the other is to be found.
That’s all from my side. Why don’t you try further from this point? Good luck!
Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Facebook page.
Problem Source: GoGeometry
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