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Understanding Isosceles Triangles and Triangle Midsegments

In this lesson, we define isosceles triangles and their properties. If a triangle is isosceles, certain relationships arise, such as congruent base angles. We will explore midsegments, which connect midpoints of a triangle's sides, and discover how to use the Triangle Midsegment Theorem to solve for missing measures. Learn to identify midsegments and apply properties to complete exercises involving the relationships within triangles. By the end, you’ll be equipped to tackle problems involving isosceles triangles and their midsegments.

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Understanding Isosceles Triangles and Triangle Midsegments

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  1. Daily Warm Up Quiz 1. Define: isosceles triangle 2. Complete each theorem below: If a triangle is isosceles, then __________. If the base angles of a triangle are congruent, then ______________________. • Find the missing measure: x

  2. Relationships Within Triangles Midsegments in Triangles

  3. Midsegments of Triangles At the end of this lesson, you should be able to use properties of midsegments to solve problems.

  4. Definition: Midsegment of a Triangle is a segment in a triangle which connects the midpoints of opposite sides. A midsegment of a triangle ______________________________________________________________________________________ Note: A is a midpoint. B is a midpoint. AB is a midsegment. 5 6 B A 6 5

  5. Midsegment of a Triangle How many midsegments are there in any triangle? 3 4 3 4 3 5 5 5 5 Three midsegments can be constructed in any triangle.

  6. Triangle Midsegment Theorem If a segment in a ∆is a midsegment, then it • is // to the third side • has length ½ the third side.

  7. Identifying Midsegments in Triangles Identify A, B, or C as a midsegment, then determine its length. B = 6 3 C = 4.5 5 5 4 3 A 5 A 7 5 5 3 B C B C 4 4 6 6 C = 7.5 A = 8 6 4 7 7 7 A A 4 7 8 6 B C B C 8 8 7 9 You can use Triangle Midsegment Theorem to determine missing measures in triangles.

  8. Using Triangle Midsegment Theorem to Find Missing Measures Q and P are the midpoints of ∆ RST. What is m RS?...m TQ?....m TS? R x + 50 P x + 85 Q ST 3x + 46 A B C 110 70 F E D Check Here! 1. Find the measure of the following: a.) ∠ ABC b.) ∠ D c.) ∠ A d.) ∠ CBF 70 70 x 40 110

  9. Final Checks: Architecture The triangular face of the Rock ‘n’ Roll Hall of Fame in Cleveland is isosceles. The length of the base is 229 ft. 6 in. What is the length of the highlighted segment?

  10. Final Checks for Understanding • Define midsegment of a triangle. • Name two properties of midsegments in triangles. • Find the missing measures, given a is a midsegment. a x x = _____; y = _____ 24 Homework: 64 26 y

  11. Check Your Solution Here: PS = x TS = 3x + 46 PS = 124 TS = 3 (124) + 46 RS = RP + PS TS = 372 + 46 RS = 124 + 124 TS = 418 RS= 248 TQ = QR TQ = x + 50 TQ = 124 + 50 TQ = 174 PQ = ½ ST x + 85 = ½ (3x + 46) 2(x + 85) = 3x + 46 2x + 170 = 3x + 46 2x = 3x – 124 -x = -124 x = 124 Click here to return to today’s lesson:

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