Isosceles Triangle Properties Proofs

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Questions and Answers

In triangle ABC, if altitudes BE and CF are equal, then triangle ABE and triangle ACF are congruent.

True (A)

What congruence criterion is used to prove that ΔABE = ΔACF?

SSS
SAS
RHS (correct)
ASA

If the altitudes BE and CF are equal in triangle ABC, what can you conclude about the sides AB and AC?

AB = AC

A triangle with two equal sides is called an ______ triangle.

<p>isosceles</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Altitude = A line segment from a vertex of a triangle perpendicular to the opposite side Congruent Triangles = Triangles that have the same size and shape Isosceles Triangle = A triangle with two equal sides RHS Congruence Rule = Two right-angled triangles are congruent if their hypotenuse and one side are equal</p> Signup and view all the answers

Flashcards

Equal Altitudes

In triangle ABC, altitudes BE and CF to AC and AB are equal.

Area of Triangle

The area of triangle ABE is equal to the area of triangle ACF.

Isosceles Triangle

Triangle ABC is isosceles; sides AB and AC are equal.

Triangle Congruence

Triangles ABE and ACF are congruent due to equal bases and heights.

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Base Comparison

If triangles ABE and ACF are congruent, their bases AB and AC are equal.

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Study Notes

Given Information

Triangle ABC has altitudes BE and CF drawn to sides AC and AB, respectively.
The altitudes BE and CF are equal in length.

Proof of ΔABE ≅ ΔACF (i)

Consider triangles ABE and ACF.
∠AEB = ∠AFC = 90° (Since BE and CF are altitudes)
∠A is common to both triangles.
BE = CF (Given)
Therefore, ΔABE and ΔACF are congruent by the right angle-hypotenuse-side (RHS) congruence criterion.