![]() Instead, what if we draw a line that bisects the apex (or top) angle:Īgain we have two triangles, ΔABD and ΔACD, where the angles we want to prove are congruent are in corresponding places. But this time, suppose you didn't think of drawing a line to the middle of the base. So what if you didn't have that intuition? Well, luckily, we can prove this in another way. I think the only "tricky" part of the above proof was the intuition required to draw the line connecting A with the middle of the base. (6) ∠ACB ≅ ∠ABC // Corresponding angles in congruent triangles (CPCTC) Another way to prove the base angles theorem Given: Angle B congruent to Angle C Prove: Segment AB congruent to Segment AC Plan for proof: Show that Segment AB and AC are corresponding parts of congruent triangles. If two angles of a triangle are congruent, then the sides opposite those angles are congruent. (4) AD = AD // Common side to both triangles Converse of the Isosceles Triangle Theorem. (3) BD = DC // We constructed D as the midpoint of the base CB (2) AB=AC // Definition of an isosceles triangle So how do we show that the triangles are congruent? Easy! Using the Side-Side-Side postulate: Proof If we show that the triangles are congruent, we are done with this geometry proof. Putting these two things together, it would make sense to create the following two triangles, by connecting A with the mid-point of the base, CB:Īnd now we have two triangles, ΔABD and ΔACD, where the angles we want to prove are congruent are in corresponding places. Then, we also want ∠ACB and ∠ABC to be in different triangles, to prove their congruency. We know that ΔABC is isosceles, which means that AB=AC, so it will be good if we place these two sides in different triangles, and already have one congruent side. So let's think about a useful way to create two triangles here. Ok, but here we only have one triangle, and to use triangle congruency we need two triangles. This is the basic strategy we will try to use in any geometry problem that requires proving that two elements (angles, sides) are equal. If we can place the two things that we want to prove are the same in corresponding places of two triangles, and then we show that the triangles are congruent, then we have shown that the corresponding elements are congruent. Triangle congruency is a useful tool for the job. This problem is typical of the kind of geometry problems that use triangle congruency as the tool for proving properties of polygons. So how do we go about proving the base angles theorem? Prove that in isosceles triangle ΔABC, the base angles ∠ACB and ∠ABC are congruent. So, here's what we'd like to prove: in an isosceles triangle, not only are the sides equal, but the base angles equal as well. ![]() We will prove most of the properties of special triangles like isosceles triangles using triangle congruency because it is a useful tool for showing that two things - two angles or two sides - are congruent if they are corresponding elements of congruent triangles. In today's lesson on proving the Converse Base Angle Theorem, we'll provide a proof for both.In this lesson, we will show you how to easily prove the Base Angles Theorem: that the base angles of an isosceles triangle are congruent. Or, draw the angle bisector of A, and use the fact that it creates a pair of equal angles at A. ![]() ![]() We can draw either the altitude to the base, and use the fact that it creates a linear pair of equal right angles. And as a result, the corresponding sides, AB and AC, will be equal.Īnd just like in the original theorem, we have a choice of which line to draw. We'll do the same here, prove the triangles are congruent relying on the fact that the base angles are congruent. As a result, the base angles were congruent. There, we drew a line from A to the base BC and proved the resulting triangles are congruent. We will try to apply the same strategy we used to prove the original one - the Base Angles Theorem. When proving the Converse Base Angle theorem, we will do what we usually do with converse theorems. In triangle ΔABC, the angles ∠ACB and ∠ABC are congruent. Now we'll prove the converse theorem - if two angles in a triangle are congruent, the triangle is isosceles. We will use congruent triangles for the proof.įrom the definition of an isosceles triangle as one in which two sides are equal, we proved the Base Angles Theorem - the angles between the equal sides and the base are congruent. In today's lesson, we will prove the converse to the Base Angle theorem - if two angles of a triangle are congruent, the triangle is isosceles.
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