Chapter 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 of the statement: “If two altitudes of a triangle are congruent, then the triangle is isosceles.” 1.Draw the shape, label everything. 2.The “if” part of the statement is the “given.” 3.The “then” part of the statement is the “prove.” 4.Write the givens and what you want to prove.

E D C B A 1.Draw a diagram to show two altitudes in a triangle. Label everything. 2.Write your given statements. 3.Write your prove statement. Given: BD & CE are altitudes to AC & AD of ACD. BD  CE. Prove: ACD is isosceles.

NOTICE!!! You can label everything on a diagram to help you make the proof. Some problems you only have to draw, label, write the givens and what to prove. Others you also have to prove.

Remember If….then…. Sometimes you will see these in reverse. “The medians of a triangle are congruent if the triangle is equilateral.” 1.Draw the diagram. 2.Write down the givens you need. 3.What do you need to prove?

1.Draw a diagram to show the medians in an equilateral triangle. Label everything. 2.Write your given statements. 3.Write your prove statement. Z Y X Q RP Given: XYZ is equilateral. PZ, RY, and QX are medians. Prove: PZ  RY  QX

One more time. Try this one! “If each pair of opposite sides of a four- sided figure are congruent, then the segments joining opposite vertices bisect each other.” 1.Draw 2.Write Given 3.Write Prove 4.Write proof

A B C D E Given: AB  CD AD  BC Prove: AC bisects BD BD bisects AC

Δ ABC  Δ CDA by SSS, and thus, <BAC  <DCA. Δ BAD  Δ DCB by SSS, and thus, <ABD  <CDB. Thus Δ ABE  Δ CDE by ASA, and then AE  EC and DE  EB.