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The Case of the Missing Diagram

4.2. The Case of the Missing Diagram. Objective. After studying this section, you will be able to: organize the information in, and draw diagrams for problems presented in words. Set up a proof for the statement:.

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The Case of the Missing Diagram

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  1. 4.2 The Case of the Missing Diagram

  2. Objective • After studying this section, you will be able to: • organize the information in, and • draw diagrams for problems presented in words.

  3. Set up a proof for the statement: “If two altitudes of a triangle are congruent, then the triangle is isosceles.” • (1)Drawthe figure,and(2)labeleverything. • The “if” part of the statement is the “given.” • The “then” part of the statement is the “prove.” • Write the (3)givens and what you want to (4)prove.

  4. 1. Draw a diagram to show two congruent altitudes in a triangle. • Next, label EVERYTHING • 3. Write your given statements. • 4. Write your prove statement. Proving: “If two altitudes of a triangle are congruent, then the triangle is isosceles.” A Given: BD & CE are altitudes to AC & AD of ΔACD BD  CE B E C D Prove: ΔACD is isosceles

  5. Remember: If (hypothesis), then (conclusion).Sometimes you will see these in the reverse. • “The medians of a triangle are congruent if the triangle is equilateral.” • 1. Draw the diagram. • 2. How will you label it? • 3. Write down the givens you have. • 4. What do you need to prove?

  6. Proving: If a triangle is equilateral, then the medians in the triangle are congruent. Think: If a triangle is EQUILATERAL, it has 3 congruent sides and is also equiangular! • Draw a diagram to show the medians in an equilateral triangle and everything else you know to be true by definition. • LabelEVERYTHING! • Write your given statements. • 4. Write your prove statement. X P R Given:ΔXYZ is equilateral. PZ, RY, and QX are medians. Y Z Q Prove: ZP  YR  XQ

  7. And if you have to PROVE that the medians are congruent . . .

  8. Remember! ALL SIDES were congruent! Don’t forget! ALL ANGLES were (60°) AND!!! If a triangle has AT LEAST two congruent sides, then ISOSCELES ! So . . . What does that mean about the MEDIANS?

  9. Sometimes a problem may not include the words “if” and “then” . . . then what? Example: Set up a proof of this statement. The opposite angles of a parallelogram are congruent. HINT • The beginning of the sentence contains the “given” information and it ends with the “conclusion”. . . but about what?

  10. PROVE:The opposite angles of a parallelogram are congruent. Either way, what do you have and how will you use it? Think… In a parallelogram, opposite sides are congruent! Given:ABCD is a parallelogram Prove: B C ?? ? ?? ? A D Hint #1: Draw AC, because two points determine a segment! Now what do you have and how will you use it? OR you could: Draw BD, because ANY two points determine a segment! B C ΔCDA ΔABC by SSS . . .And now what else is provable? C Final HINT! A D A

  11. 4.2 Missing Diagram Partner Problems(Notes packet page 2) • Work with your table partner on problems 1 - 4. • Set up the proof: • Draw the diagram • Label everything • Write the “Givens” • Write the “Prove” DO NOT SOLVE!!! • Try not to spend more than 5 minutes on each problem.

  12. 4.2 Homework Pp. 178 - 179 • 5- 7 (set up only), • 9 & 10 (set up and write proofs)

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