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Chapter 8: Problem 14

Prove that if the vertex angle of one isosceles triangle is congruent to the vertex angle of a second isosceles triangle, the triangles are similar.

Short Answer

Expert verified
Two isosceles triangles whose vertex angles are congruent are similar, as demonstrated by the base angles and the Angle-Angle (AA) similarity rule. The base angles in each isosceles triangle are congruent (a property of isosceles triangles), and the congruence of the vertex angle (given in the problem) completes the two pairs of equivalent angles required for similarity by AA.

Step by step solution

01

Identify the key properties of isosceles triangle

An isosceles triangle is a triangle that has two sides of equal length and the angles opposite those sides are equal. Therefore, each isosceles triangle has at least two congruent angles.
02

Apply Angle-Angle Rule

According to the Angle-Angle (AA) similarity rule, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
03

Prove the similarity of two isosceles triangles

Given that the vertex angle of one isosceles triangle is congruent to the vertex angle of a second isosceles triangle, it indicates that one angle of the first triangle is equal to one angle of second triangle. Combined with the base angles, which are congruent by definition of an isosceles triangle, we have two pairs of congruent angles thus proving by the AA rule that the triangles are similar.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isosceles Triangle
An isosceles triangle is a special type of triangle that has some unique characteristics. The most notable feature of an isosceles triangle is that it has two sides of equal length. These equal sides are commonly referred to as the "legs" of the triangle. Additionally, the angles opposite these equal sides are also equal. These are called the "base angles."
In every isosceles triangle, the third side is called the "base," and the angle opposite the base is called the "vertex angle." Understanding these features is fundamental when working with these triangles, especially in proving properties like similarity. Remember:
  • The two equal sides are always opposite two equal angles.
  • The vertex angle is not equal to the base angles unless the triangle is also equilateral.
Angle-Angle Rule
The Angle-Angle Rule, often abbreviated as the AA rule, is an essential concept when it comes to proving triangle similarity. This rule states that two triangles are similar if two angles of one triangle are congruent to two angles of another triangle. The similarity of triangles implies that their corresponding sides are proportional, and they have the same shape but potentially different sizes.
It's important to note:
  • Only two pairs of angles need to be congruent because the third pair will automatically be congruent due to the triangle angle sum property (sum of angles in a triangle is always 180掳).
  • This rule is extremely powerful and often simplifies many geometric proofs involving triangles.
Congruent Angles
Congruent angles are angles that have exactly the same measure. In geometry, recognizing congruent angles is fundamental in proving many properties about shapes, including triangle similarity. When two angles in one triangle are congruent to two angles in another triangle, this lays the foundation for using the Angle-Angle Rule to prove similarity.
In the context of isosceles triangles:
  • The base angles of an isosceles triangle are always congruent by definition.
  • If the vertex angles of two different isosceles triangles are also congruent, then by having two pairs of congruent angles (base and vertex), we can conclude the triangles are similar.
Understanding and identifying congruent angles is key to solving many geometric problems.

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Most popular questions from this chapter

Prove that if an acute angle of one right triangle is congruent to an acute angle of another right triangle, the triangles are similar. (GRAPH CANT COPY)

If \(x(a+b)=y(c+d),\) find the ratio of \(x\) to \(y\)

Find the positive arithmetic and geometric means between each pair of numbers. Note which mean is greater in each case. a 8 and 50 b 6 and 12

$$\begin{aligned} &\text { Given: } A=(1,2), B=(9,8), C=(1,8)\\\ &\mathrm{P}=(5,-3), \mathrm{Q}=(-7,6), \mathrm{R}=(-7,-3)\\\ &\mathrm{AB}=10, \mathrm{PQ}=15\\\ &\text { By which theorem is } \triangle \mathrm{ABC} \sim \triangle \mathrm{QPR} ? \end{aligned}$$

Prove that the ratio of corresponding altitudes of similar triangles is equal to the ratio of any pair of corresponding sides of the triangles.
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