Mathematics
Grade 10
15 min
Proofs involving isosceles triangles
Proofs involving isosceles triangles
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Introduction & Learning Objectives
Learning Objectives
Identify the properties of an isosceles triangle, including legs, base, vertex angle, and base angles.
State and apply the Isosceles Triangle Theorem and its converse in proofs.
Construct a formal two-column proof to demonstrate relationships within a single isosceles triangle.
Use the properties of isosceles triangles to prove two triangles are congruent.
Apply CPCTC (Corresponding Parts of Congruent Triangles are Congruent) after establishing triangle congruence involving isosceles triangles.
Identify and use auxiliary lines, such as medians and angle bisectors, to solve proofs involving isosceles triangles.
Have you ever noticed the strong, symmetrical shape of an A-frame house or a suspension bridge? 🏛️ That powerful symmetry comes from isosceles tria...
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Key Concepts & Vocabulary
TermDefinitionExample
Isosceles TriangleA triangle with at least two congruent sides.In ΔABC, if side AB is 5 cm and side AC is 5 cm, then ΔABC is an isosceles triangle.
Legs of an Isosceles TriangleThe two congruent sides of an isosceles triangle.In isosceles ΔABC with AB ≅ AC, the legs are segments AB and AC.
Base of an Isosceles TriangleThe side opposite the vertex angle. It is the side that is not necessarily congruent to the other two.In isosceles ΔABC with AB ≅ AC, the base is segment BC.
Vertex AngleThe angle formed by the two congruent legs of an isosceles triangle.In isosceles ΔABC with AB ≅ AC, the vertex angle is ∠BAC.
Base AnglesThe two angles that are adjacent to the base of an isosceles triangle. These angles are opposite the congruent legs.In isosceles ΔABC with AB ≅ AC, th...
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Core Formulas
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Use this theorem when you are given that a triangle has two equal sides and you need to prove that its base angles are equal. In a proof, if you know $\overline{AB} \cong \overline{AC}$, you can conclude that $\angle C \cong \angle B$.
Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Use this theorem when you are given that a triangle has two equal angles and you need to prove that the sides opposite them are equal. In a proof, if you know $\angle C \cong \angle B$, you can conclude that $\overline{AB} \cong \overline{AC}$.
Properties of the Altitude to the Base...
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Challenging
Given: Isosceles ΔABC with AB ≅ AC. D is the midpoint of AB, and E is the midpoint of AC. Which of the following is the most logical final conclusion after proving ΔABE ≅ ΔACD?
A.BE ≅ CD
B.AD ≅ AE
C.∠AEB ≅ ∠ADC
D.BC ≅ BC
Challenging
In isosceles ΔABC with base BC, an altitude CD is drawn from base angle C to the leg AB. Which of the following statements is NOT necessarily true?
A.ΔADC is a right triangle.
B.CD bisects ∠ACB.
C.Another altitude, BE, from B to AC would be congruent to CD.
D.The area of ΔABC is (1/2)(AB)(CD).
Challenging
You are asked to prove that if the median to a side of a triangle is also the altitude to that side, then the triangle is isosceles. Given median AM is also altitude AM in ΔABC, what is the key step to prove AB ≅ AC?
A.Prove ΔABM ≅ ΔACM by SSS.
B.Prove ΔABM ≅ ΔACM by ASA.
C.Prove ΔABM ≅ ΔACM by SAS.
D.Use the Converse of the Isosceles Triangle Theorem on ΔABC.
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