4.7 Isosceles and Equilateral TrianglesObjective: You will use theorems about isosceles and equilateral triangles.

Base Angle TheoremIf two sides of a triangle are congruent, then the angles opposite them are congruent.

Converse of Base Angles TheoremIf two angles of a triangle are congruent, then the sides opposite them are congruent.

CorollaryCorollary to the Base Angle Theorem- if a triangle is equilateral, then it is equiangular.

Corollary to the Converse of Base Angle Theorem- If a triangle is equiangular, then it is equilateral.

EXAMPLE 1Apply the Base Angles TheoremSOLUTIONIn DEF, DE DF . Name two congruent angles.

GUIDED PRACTICEfor Example 1SOLUTION

GUIDED PRACTICEfor Example 1SOLUTION

EXAMPLE 2Find measures in a triangle

GUIDED PRACTICEfor Example 2SOLUTION

GUIDED PRACTICEfor Example 2SOLUTIONNo; it is not possible for an equilateral triangle to have angle measure other then 60. Because the triangle sum theorem and the fact that the triangle is equilateral guarantees the angle measure 60 because all pairs of angles could be considered base of an isosceles triangle

EXAMPLE 3Use isosceles and equilateral trianglesSOLUTION

EXAMPLE 3Use isosceles and equilateral trianglesLN = LMDefinition of congruent segments4 = x + 1Substitute 4 for LN and x + 1 for LM.3 = xSubtract 1 from each side.

EXAMPLE 4Solve a multi-step problemLifeguard Tower

EXAMPLE 4Solve a multi-step problemSOLUTION

EXAMPLE 4Solve a multi-step problem You know that PS QR , and 3 4 because corresp. parts of are . Also, PTS QTR by the Vertical Angles Congruence Theorem. So, PTS QTR by the AAS Congruence Theorem.

GUIDED PRACTICEfor Examples 3 and 4SOLUTION

GUIDED PRACTICEfor Examples 3 and 4SOLUTION

GUIDED PRACTICEfor Examples 3 and 4Since PT QT from part (b) andfrom part (c) then,

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