![]() We can then find the values of □ and □ by solving these two equations simultaneously either by substitution or elimination. Adding one to both sides gives us a second equation of five □ is equal to 11 minus three □. This gives five □ minus one is equal to 10 minus three □. For the second equation, we can fill in the lengths for □□ and □□. We can’t solve this just yet as there are two unknowns, so let’s label this with equation one. We can rearrange this by subtracting three from both sides to give us three □ minus one is equal to five □. This would give us three □ plus two is equal to five □ plus three. So let’s take this first equation □□ is equal to □□ and fill in the given expressions for these lengths. And if line segment □□ is bisected, then □□ is equal to □□. So when this is an isosceles triangle, the two legs □□ and □□ will be congruent. That means that the line segment □□ is only a perpendicular bisector of line segment □□ in the case of an isosceles triangle. And in this question, we are interested in the perpendicular bisector of the line segment □□, which would be the base of this triangle. We can recall that an isosceles triangle is a triangle that has two congruent sides and the median of an isosceles triangle from the vertex angle is a perpendicular bisector of the base. Although we can’t prove this, we can use some of the properties of isosceles triangles to help. In the figure, we can observe that triangle □□□ appears to be an isosceles triangle. We’ll now see how we can apply this corollary in the following example.įor which values of □ and □ is line segment □□ a perpendicular bisector of line segment □□? Because of this corollary, we can also notice that a useful property of the median of an isosceles triangle is that it forms the axis of symmetry of this triangle because it splits the isosceles triangle into two congruent right triangles. The median of an isosceles triangle from the vertex angle bisects it and is perpendicular to the base. Furthermore, since the line segment □□ is a straight line, then both of these angles □□□ and □□□ must be 90 degrees. That means that the measure of angle □□□ is congruent to the measure of angle □□□. We can also write that □□ is equal to □□ because we know that the triangle is isosceles and the line segment □□ is a shared side in the two triangles □□□ and □□□.Īnd as there are now three congruent pairs of sides, then we can say that triangle □□□ is congruent to triangle □□□ by the SSS, or side-side-side, congurrency criterion. We can therefore say that □□ is congruent to □□. The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side and therefore bisecting that side. Let’s take this isosceles triangle □□□, and we draw the median from □ to create the point □. We can prove this corollary in the following way. This corollary states that the median of an isosceles triangle from the vertex angle bisects it and is perpendicular to the base. Let’s see the first of these corollaries. These corollaries will allow us to identify additional geometric properties about isosceles triangles. We will now consider a number of corollaries to these theorems. That is, if two angles of a triangle are congruent, then the sides opposite those angles are also congruent. And the converse of this theorem is also true. And by this theorem, it means that they also have two congruent angles. So we know that isosceles triangles by definition have two congruent sides. The remaining angle in the isosceles triangle is referred to as the vertex angle. So knowing that the two sides are congruent means that in fact the two base angles are congruent. It is the isosceles triangle theorem, and it states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. ![]() Now, because isosceles triangles have two congruent sides, this leads us to an important angle property of isosceles triangles. The congruent sides are called the legs of the triangle and the third side is the base. And when we are talking about isosceles triangles, we use two important terms. We can recall that an isosceles triangle is simply a triangle that has two congruent sides. In this video, we will learn how to use the corollaries of the isosceles triangle theorems to find missing lengths and angles in isosceles triangles.
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