Exploring Isosceles Triangles: Properties and Theorems
Discover the fascinating world of isosceles triangles! This guide covers essential properties, including the Isosceles Triangle Theorem, which states that if two sides are congruent, the angles opposite them are also congruent. Engage in practical activities by drawing acute and obtuse isosceles triangles, cutting them out, and folding them to explore their characteristics. Learn how to use algebra to find missing measures related to isosceles triangles. Deepen your understanding of geometry with proofs, corollaries, and the relationships between sides and angles in triangles.
Exploring Isosceles Triangles: Properties and Theorems
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Presentation Transcript
Honors Geometry Isosceles Triangles
Remember– the properties of an isosceles triangle….. Vertex Angle Vertex Angle Leg Leg BaseAngles Base
Investigating Isosceles Triangles • Use a straightedge to draw an ACUTE ISOSCELES triangle-- where and is the acute vertex angle. • Use scissors to cut the triangle out • Then fold the triangle as shown • REPEAT the procedure for an OBTUSE ISOSCELES triangle -- where and is the obtuse vertex angle. • What observation can you make about the base angles?
Isosceles Triangle Theorem • If two sides of a triangle are congruent, then the angles opposite them are congruent.
Use ALGEBRA to find the missing measures(not drawn to scale) • 1. 44 m r 30 x y
Use ALGEBRA to find the missing measures(not drawn to scale) • 1. • x+y+ 44 = 180 Sum • x= y because the two base angles are congruent to each other b/c they are opposite congruent sides • 180 = x + x + 44 • 136 = 2x • 68=x • 68 = y 44 m r 30 68 68 x y
Use ALGEBRA to find the missing measures(not drawn to scale) • 2. m r 30°
Find the missing measures(not drawn to scale) • 30 + r + m = 180 • r is the other base angle and must be 30° b/c its opposite from a congruent side. • 30 + 30 + m = 180 • 60 + m = 180 • m = 120 • 2. m 120° r 30° 30°
Isosceles Triangle Theorem • If two sides of a triangle are congruent, then the angles opposite them are congruent. • Given: • Prove:
Proof of Base Angles Theorem • Statements • Label H as the midpoint of CY • Draw NH • Reasons • Ruler Postulate • 2 points determine a line • Def. of midpoint • Reflexive Prop • Given • SSS • CPCTC • Given: Prove:
Converse of the Isosceles Triangle Theorem • If two angles of a triangle are congruent, then the sides opposite them are congruent. R A T
Corollary-- • A corollary is a theorem that follows easily from a theorem that has already been prove. • Corollary : If triangle is equilateral, then it is also equiangular. A BC • Corollary : If a triangle is equiangular, then it is also equilateral. W • E R
