1. SSS (side, side, side)
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1 CONGRUNT TRIANGLS If one shape can become another using Turns, Flips and/or Slides, then the shapes are Congruent ROTATION / TURN RFLCTION / FLIP TRANSLATION/SLID After any of those transformation (turn, flip or slide), the shape still has the same size, area, angles, and line lengths. Two triangles are congruent if they have: How to Find if Triangles are Congruent actly the same three sides and eactly the same three angles. But we don t have to know all three sides and all three angles usually three out of the si is enough. There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS, and HL. 1. SSS (side, side, side) SSS stands for "side, side, side" and means that we have two triangles with all three sides equal. is congruent 2. SAS (side, angle, side) SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal. is congruent Page 1 of 6
2 3. ASA (angle, side, angle) ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal. 4. AAS (angle, angle, side) is congruent AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal. is congruent 5. HL (hypotenuse, leg) HL stands for Hypotenuse, Leg. This one applies only to right angled-triangles! is congruent to MARKING When two triangles are congruent we often mark corresponding sides and angles like this: ΔABC ΔRST Page 2 of 6
3 XAMPL #1: Are these triangles congruent? Which congruence sufficiency condition applies? 6 cm A 6cm D 12 cm a) State the congruency statement. i.e. Δ Δ B 8 cm 12 cm 8 cm F C XAMPL #2: ΔNPQ ΔRST. State the values of, y, and z. XAMPL #3: ΔFG ΔHJK. State the values of, y, and z. 80 K 6 cm 8.5 cm z F 9.6 cm G H XAMPL #4: Show that these triangles are congruent. State the congruency statement, the sufficiency condition, and state all evidence. 8 cm A a) Determine the values of, y, and z. z 100 b) Determine the values of, y, and z. y 62 J y 5.2 cm B D 40 C Page 3 of 6
4 Similar triangles have: all their angles equal corresponding sides have the same ratio These triangles are SIMILAR. SIMILAR TRIANGLS (qual angles have been marked with the same number of arcs) If one shape can become another using Resizing (dilation, contraction, compression, enlargement) then these Shapes are Similar! How to Find if Triangles are Similar But we don't need to know all three sides and all three angles...two or three out of the si is usually enough. There are three ways to find if two triangles are similar: AA, SAS and SSS: AA AA stands for "angle, angle" and means that the triangles have two of their angles equal. If two triangles have two of their angles equal, the triangles are similar. ample: SAS SAS stands for "side, angle, side" and means that we have two triangles where: the ratio between two sides is the same as the ratio between another two sides and we we also know the included angles are equal. ample In this eample we can see that: one pair of sides is in the ratio of 21 : 14 = 3 : 2 another pair of sides is in the ratio of 15 : 10 = 3 : 2 there is a matching angle of 75 in between them If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar. Page 4 of 6
5 SSS SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio. ample: If two triangles have three pairs of sides in the same ratio, then the triangles are similar. CORRSPONDING SIDS In similar triangles, the sides facing the equal angles are always in the same ratio. Triangles R and S are similar. The equal angles are marked with the same numbers of arcs. What are the corresponding lengths? The lengths 7 and a are corresponding (they face the angle marked with one arc) The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs) The lengths 6 and b are corresponding (they face the angle marked with three arcs) CALCULATING TH LNGTHS OF CORRSPONDING SIDS It may be possible to calculate lengths we don't know yet. We need Step 1: Find the ratio of corresponding sides in pairs of similar triangles. Step 2: Use that ratio to find the unknown lengths. 1: Find lengths a and b of Triangle S. Page 5 of 6
6 ample 1: a. plain why ΔABC ~ ΔDF. Justify your answer. B F 5 a 9 A o 7 C e b. Determine the values of a and e. 10 o D ample 2: Are the two triangles in the diagram similar? plain your reasoning. Page 334 # 7, 8 Page 6 Page 348 # 5, 6, 7, 9
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