## Book I. Proposition 26Problems Back to Proposition 26. 1. Name three sets of sufficient conditions for triangles to be congruent. To see the answer, pass your mouse over the colored area. Side-angle-side. Side-side-side. Angle-side-angle. Or equivalently, Side-angle-angle. 2. a) State the hypothesis of Proposition 26. Two triangles have two angles equal to two angles respectively, and one side equal to one side, which may be either the sides between the equal angles or the sides opposite one of them. 2. b) State the conclusion. The remaining sides will equal the remaining sides (namely those that are oppostie the equal angles), and the remaining angle will equal the remaining angle. 2. c) Prove the second case in which the sides opposite one of the 2. c) That is, let angles B and C be equal respectively to angles E and F,
If they are not equal, then assume that BC is greater; make BH equal to EF, and draw AH. 3. In quadrilateral ABCD, angles CDB, DBA are equal, and angles 3. ADB, DBC are equal. Prove that AD is equal to BC.
Since angles ADB, DBA are equal respectively to angles DBC, CDB, 4. In the rectangle ABCD, angle ABD is equal to angle BDC. 4. Prove that angle ADB is equal to angle DBC. 4. (A rectangle is a quadrilateral in which all the angles are right angles.)
The right angle at A is equal to the right angle at C, 5. In this figure, the angles at B and C are right angles, the straight line
Angle BDA is equal to angle CDE; (I. 15) 6. Use Proposition 26 to prove Proposition 6 directly: 6. 6. In triangle ABC, let angle B equal angle C; then side AB is equal (
Angle B is equal to angle C, (Hypothesis) The following problems will depend on proving congruence; either S.A.S., S.S.S., A.S.A. or S.A.A. 7. Prove: 7. (That is, it is the perpendicular bisector of the base.)
Let triangle ABC be isosceles with side AB equal to side AC; 8. In quadrilateral ABCD, the straight line AC is the perpendicular
9. a) Prove that triangles ABD and BCD are both isosce1es.
BE is equal to ED; (Hypothesis) 9. b) Prove that angle ABC is equal to angle ADC.
Since triangles DAB, BCD are isosceles, 19. BDEC is a straight line, AB is equal to AC, and AD is equal to AE. 19. Prove that BD is equal to EC.
We will show that triangles ADB, AEC are congruent. First, 10. Angles EBA and CBD are right angles. EB is equal to BA, and DB is equal to BC. Prove that triangles EBC, ABD are congruent.
Angle EBA is equal to angle CBD Table of Contents | Introduction | Home Please make a donation to keep TheMathPage online. Copyright © 2012 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |