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Chapter 7: Problem 112

Prove that the altitudes drawn to the legs of an isosceles triangle are congruent.

Short Answer

Expert verified
In an isosceles triangle with equal sides "a" and base "b", we draw altitudes h_1 and h_2 from the vertices opposite the legs. We consider the exterior triangles formed by each altitude and half of the base. Using Side-Angle-Side (SAS) congruence, we prove that these exterior triangles are congruent. Then, by applying the Converse of the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, we find that h_1 and h_2 are congruent. Therefore, the altitudes drawn to the legs of an isosceles triangle are congruent.

Step by step solution

01

Draw the Isosceles Triangle and Altitudes

Begin by drawing an isosceles triangle, with equal sides labeled as "a" and the base labeled as "b". The altitude of this triangle is the perpendicular line drawn from the vertex opposite the base to the base itself. Draw the two altitudes from the two vertices opposite legs of the isosceles triangle and label their lengths as "h_1" and "h_2".
02

Prove that the Two Altitudes are Congruent

Now we will prove that h_1 and h_2 are congruent. Consider the triangles formed by each altitude and half of the base of the isosceles triangle (from dividing the base by the altitude). By using the property of an isosceles triangle, we have the congruence of the two exterior triangles: 1. Side-Angle-Side (SAS) congruence: In these two exterior triangles, the angles are formed right angles from the altitudes, and the sides "a" are equal. 2. Using SAS, we can prove that these two triangles are congruent.
03

Apply the Converse of the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem

As we have already proved that the two exterior triangles are congruent, we can now use the Converse of the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem. According to the CPCTC: If two triangles are proven to be congruent, their corresponding parts are also congruent. Hence, in this case, the corresponding parts are the altitudes of the two triangles, h_1 and h_2.
04

Conclude that the Altitudes are Congruent

By applying the CPCTC theorem, we have now proven that the altitudes h_1 and h_2 of the exterior triangles are congruent. Therefore, the altitudes drawn to the legs of an isosceles triangle are congruent.

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Key Concepts

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

Altitudes
In geometry, an altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side. In an isosceles triangle, altitudes play a crucial role in various proofs and theorems. In our case, we explore the altitudes drawn from the vertices opposite the equal sides of an isosceles triangle. These altitudes intersect the base, dividing it into two equal parts. This creates two right-angled triangles.
  • Each of these altitudes, denoted as "h_1" and "h_2", forms right angles (90 degrees) with the base.
  • This perpendicular relationship is a foundational aspect when proving congruence using geometry, benefiting from right triangles鈥� properties.
Understanding altitudes is essential because they offer symmetry and consistency in isosceles triangles, making them easier to analyze mathematically.
Congruence
Congruence in triangles is the idea that two triangles are identical in shape and size, even if their positions or orientations differ. This concept is fundamental when dealing with triangles because it allows us to establish equality between various parts of different triangles.
  • In isosceles triangles, congruence is used to show that the altitudes "h_1" and "h_2" are equal in length.
  • If two triangles are congruent, all their corresponding interior angles and sides are equal.
When we prove congruence between triangles, like in our isosceles triangle, we can use congruent parts to make deductions about the lengths of altitudes, which is the central proof goal here.
Side-Angle-Side (SAS)
The Side-Angle-Side (SAS) criterion is one of several methods used to prove the congruence of two triangles. The SAS postulate states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
  • In this exercise, the sides "a" and the right angle formed by the altitudes of the two triangles provide the basis for SAS.
  • As each altitude is perpendicular to the base, forming a right angle, and the sides "a" (equal sides of the isosceles triangle) are shared, the SAS postulate confirms congruence.
Utilizing SAS is crucial because it provides the initial step needed to show that the resulting smaller triangles in our isosceles shape have identical characteristics.
Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem is an essential idea that follows from proving triangle congruence. It states that if two triangles are proven to be congruent, then each of their corresponding parts (angles and sides) are congruent too.
  • Once the triangles formed by the altitudes are proven congruent through SAS, CPCTC is applied to conclude that the altitudes themselves are equal in length.
  • This logical step validates our proof of the altitudes being congruent, as it directly stems from already-proven congruent triangles.
By connecting congruence and CPCTC, we can seamlessly transition from proving triangles congruent to proving intricate parts within them are equal, further demonstrating the elegant structure of geometric reasoning.

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Most popular questions from this chapter

Let \(\triangle \mathrm{ABC}\) be a triangle such that \(\angle \mathrm{B} \cong \angle \mathrm{C}\). Use the ASA Theorem to prove that \(\mathrm{AB}=\mathrm{AC}\). (Don't use the Isosceles Triangle Theorem.)

Given \(\triangle \mathrm{ABC}, \underline{\mathrm{AC}} \cong \underline{\mathrm{BC}}, \angle 1 \cong \angle 2 .\) Prove: \(\angle 3 \cong \angle 4\)

Given: \(\underline{A E}\) and \(\underline{B D}\) are straight lines intersecting at C. \(\underline{B C}\) \(\cong \underline{D C} ; \angle B \cong \angle D\). Prove: \(\Delta A B C \cong \triangle E D C\).

Given \(\angle 1 \cong \angle 2 ; \angle 3 \cong \angle 4\). Prove \(\mathrm{RM}=\mathrm{RN}\).

If \(\angle 3 \cong \angle 4\) and \(Q \underline{M}\) bisects \(\angle P Q R\), prove that \(M\) is the midpoint of \(\underline{P R}\).
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