Chapter 9 Congruence, Symmetry and Similarity Section 9.3 Constructions and Congruence

In the last section we ended by making the comment that if all the corresponding parts of two shapes were congruent then so are the two shapes. In this section we show that in a triangle all of the parts (angles and sides) being congruent is not required, we can get by with only some of them if they are arranged in certain patterns. Compass and Straightedge The “tools” we use to copy parts of a triangle are a compass and a straightedge. A compass is a device used to draw circles or parts of circles called arcs. A straightedge is like a ruler but with no markings on it. A ruler or yard stick is often used but you must ignore the markings. Copying a segment The compass and straightedge can be used together to transfer a segment of a given length onto a line. This is done in two steps: 1. Put point on A and open till mark is on B 2. Lift off and put point on C and mark point D B A C D

Side-Side-Side (SSS) Triangle Congruence If three sides of a triangle are congruent to the three corresponding sides of another triangle, then the two triangles are congruent. We show this by showing how segments from one triangle can be translated (copied by a compass and straightedge) to form the other triangle. A D F B C E Steps to copy a triangle by coping the sides: 1. Copy segment to locate point F 2. Make arc of length with point at F 3. Copy segment to locate point D on the arc from step (2) 4. Use straightedge to fill in segment Now we have, ABC  DEF

Copying an Angle Copying an angle can be accomplished by copying a triangle that is included in that angle. A B C Steps to copy an angle: 1. Swing arc on the original angle (ABC) and without changing it make same arc on the other ray you want to copy it onto. 2. Make arc from where the arc in step (1) passed through the original angle and transfer it to the ray you want to copy it onto. 3. With your straightedge draw the line that connect the endpoint and where the arcs cross.

Side-Angle-Side (SAS) Triangle Congruence If two sides and the included angle of one triangle are congruent to two corresponding sides and the included angle of another triangle with the corresponding sides being congruent, then the triangles are congruent. The included angle of two sides of a triangle is the angle that is formed by the two sides of the triangle. It can not just be any two congruent sides and an angle, but the angle that is between the two sides. Below we show how to use a compass and straightedge to copy the side-angle-side of a triangle. A Steps to copy a triangle by copying a side-angle-side: 1. Copy ABC with vertex at point E. 2. Use straightedge to draw in 3. Copy onto 4. Copy onto 5. Draw segment 6. ABC  DEF B C D E F

A compass and straightedge can be used to construct both angle bisectors and perpendicular bisectors of segments. Angle Bisector Perpendicular Bisector Base Angles of Isosceles Triangles are Congruent In an isosceles triangle the angles made with the non-congruent side and one of the congruent sides are called the base angles. In the triangle to the right ABD and ACD are the base angles. The base angles are congruent. The reason for this is as follows: 1. Construct angle bisector for CAB and call the point of intersection with point D. 2. BAD  CAD (Side-Angle-Side) 3. ABD  ACD (They are the corresponding parts of the congruent triangles.) A B D C

The angles v and x are supplementary. y = 20 and z = 20 The picture to the right is a circle with center point A and a triangle inscribed in it. Inscribed in a circle means all the vertices of the triangle are on the circle. The angle that is on the bottom right of the triangle is 70. Use what you know about congruent angles and the measure of the interior angles of triangles to determine what the measures of all the other angles are. D y u z x v 70 C B A u = 70 The key here is to realize DAB is isosceles since two of the sides are radii of the circle and u and 70 are base angles. v = 40 The sum of interior angles of a triangle are 180. v+u+70=v+140=180 x = 140 The angles v and x are supplementary. y = 20 and z = 20 This is the hardest. The angles y and z have the same measure since they are both base angles of an isosceles triangle. The sum of interior angles of a triangle are 180. y+z+x = y+y+160 = 2y+160=180