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How to construct congruent triangles using three sides (SSS) Using a straight edge construct a scalene triangle and label all the vertexes (ex: ΔABC), draw an image ray and label the endpoint (ex: X) Then (SSS triangle construction) Given three sides (ex AB - longest, BC - middle, and CA - shortest) (a) (b) (c) (d) (e) Copy the longest segment onto the image ray labeling the new endpoint and mark the congruent parts with “Tic marks” (ex if AB is the longest side copy AB to ray “X” and label the new endpoint “Y” - and put a single tic mark on each segment) Span a radius equal to the length of middle length side and copy it to one endpoint on the image ray. Draw the arc such that it is towards the other endpoint but not thru the segment XY. (ex. Radius BC to image ray endpoint X) Span a radius equal to the length of shortest side and copy it to the other endpoint on the image ray. Draw the arc such that it intercepts the arc drawn in step (b) above (ex: Radius CA to image ray endpoint Y) Where the two arcs from steps (b) and (c) intersect is your third point Z. Label it Z and connect it to X and Y, and put double and triple tic marks on the original and image sides to show congruence. Box a triangle congruence statement How to construct congruent triangles using two sides and the included angle (SAS) or two angles and the included side (ASA) Using a straight edge construct a scalene triangle and label all the vertexes (ex: ΔDEF), draw an image ray and label the endpoint (ex: R) Then (SAS triangle construction) Always copy the part represented by the middle letter first (and pick the largest one) Given angle B is the largest angle (and CA is the longest side) (a) Copy the largest angle onto the ray. Hint: Use the length of one of your shortest side (ex: AB).as your measuring arc, (b) After copying your angle. Label the one side you used for the measuring arc (you can pick either side of your angle). Put tic marks for the angle and the side to show congruence to the originals (ex: new point T, congruent sides BA to RT, and angles B to R) (c) Span the other side of your original angle B (ex: BC) and copy it to the other side of your image angle R, and label point and congruence (ex: new point S, and congruent sides BC to RS) (d) Connect the two new points to complete the triangle, and box a triangle congruence statement. Using a straight edge construct a scalene triangle and label all the vertexes (ex: ΔGHI), draw an image ray and label the endpoint (ex: “J”) Then (ASA triangle construction) Given side IG is the longest side (and angle H is the largest angle) (a) Copy the longest side onto the image ray labeling the new endpoint (ex: “K”). (b) Copy the both angles from the endpoints of the longest side to the endpoints on the images side. Hint: You can use the same measuring ray for both angles. Therefore you make a total four arcs with the same radius (two on your originals, and two on the endpoints for your image side.) (c) After copying your angles, put tic marks for the angles to show congruence to the originals. Also mark the congruence for the included side made in step (a). Extend the sides to meet at your third vertex. (d) The intersection of the new sides is your third point. Label it (ex “L”) to complete the triangle. (e) Box a congruence statement How to construct triangles using Hypotenuse Leg (HL) as your originals parts The first part is to construct a right angle for your original triangle using these steps: (1) Draw a line with a point near the midpoint, R (2) Construct a semicircle about this point R that passes thru the line twice (3) From the two intersections of this semicircle and the line, construct two arcs that intersect on one side of the line (4) Draw a line thru the intersection of the two arcs in step (3) thru the point R (HL triangle construction) Given two sides one identified as a Hypotenuse and the other as a Leg and (ex Hyp: AB, Leg: BC) (a) First you need to figure out which letter corresponds to your right angle. It is always the one letter NOT on the hypotenuse. In the example the angle Hypotenuses is AB, so the missing letter is the other letter in the leg – C (b) On your image ray you need to pull a right angle from the point you found in step (a). When doing this construction, use the leg length as your semicircle, then label your right angle with a “box” and letter, and then the leg with the other endpoint and make the tic marks on the leg to agree with that on your original. For the example: right angle is C’, and the leg has B’ at the other endpoint. (c) Span the hypotenuse, and place the bulls-eye over the endpoint of the leg in step (b). Make an arc to pass thru the other side of your right angle. Note: you may need to extend that other side long enough so it passes thru the arc. Where the arc passes the side label the last point and make the tic marks on the hypotenuse to agree with the original (d) Box a congruence statement