Chapter 3: Congruent to Similar Figures (Triangles) Angles. Opposite Angles: Corresponding Angles: Alternate Interior Angles

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1 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 1 hapter 3: ongruent to Similar Figures (Triangles) ngles Name the following angles: omplementary ngles Supplementary ngles 70 o 140 o Opposite ngles: orresponding ngles: lternate Interior ngles

2 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 2 ongruent Triangles ( ) ongruent Triangles are triangles whose corresponding angles and corresponding sides are congruent. If triangles are congruent, they have the exact same side lengths and angles. There are 3 ways to prove that triangles are congruent: 1) SSS Side-Side-Side If all of the corresponding side lengths are the same in two triangles, the triangles are congruent. 2) S ngle-side-ngle If two angles are congruent, and the side between them is congruent in two triangles, the triangles are congruent. 3) SS Side-ngle-Side If two corresponding sides are congruent, and the angle between them is congruent in two triangles, the triangles are congruent.

3 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 3 Similar Triangles (~) Similar Triangles are triangles whose corresponding sides are congruent and whose corresponding sides are proportional in length. There are 3 ways to prove that triangles are similar: 1) ngle-ngle If two corresponding angles are congruent in two triangles, the triangles are similar. 2) SS Side-ngle-Side If two corresponding sides are proportional, and the angle between them is congruent, in two triangles, the triangles are similar. 3) SSS Side-Side-Side If all three corresponding sides are proportional in two triangles, then the triangles are similar. If you know that triangles are similar, it is possible to find missing measurements. If and TUV are similar, what is the measurement of side UV? 2 cm T 11.2 cm 4.5 cm U? V

4 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 4 Metric Relations in a Right Triangle Steps to solve Metric Relation Problems a m h c n b 1. Identify the problem as metric relations (must be a right triangle with the height drawn in) 2. Label the diagram a & m on the same side b & n on the same side 3. What do you know? What are you looking for? 4. Select appropriate formula 5. Plug in values and solve a 2 = m c b 2 = n c h 2 = m n a b = c h Find the missing values in the following triangles: a)

5 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 5 hapter 3 Practice Questions 1. onsider the two triangles below. Which geometric statement proves that E D? ) SS ) S E D ) SSS D) 2. In the figure to the right, segments DE and are parallel and they measure 6 units and 10 units respectively. Segment E measures 3 units.? E D What is the measure of segment E? ) 2 units ) 4.5 units ) 5 units D) 7.5 units 3. statue, honoring Ray Hnatyshyn ( ), can be found on Spadina rescent East, near the University ridge in Saskatoon. Use the information below to determine the unknown height of the statue.

6 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 6 4. Triangles EFG and QRS are similar. The length of the sides of EFG are 144, 128, and 112. The length of the smallest side of QRS is 280, what is the length of the longest side of QRS? (draw a diagram and solve) 5. girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How high is the lamp post? 160 cm 90 cm 360 cm 6. Triangle STU shown to the right is right-angled at T. In addition: S What is the perimeter of triangle STU? 16 cm W U 36 cm T

7 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 7 7. Find the distance across the river in the sketch below. X?? The River Y 64 m Z 20 m W 16 m T 8. Given triangle with a right angle at. D is drawn perpendicular to at D and DE is drawn perpendicular to at E. The height D measures 12 cm, hypotenuse measures 25 cm and side measures 20 cm. E D Find the measure of DE.

8 h. 3 n g l e s, T r i a n g l e s a n d M e t r i c R e l a t i o n s P a g e 8 hapter 3 nswer Sheet hapter 3 Practice Questions D 3. 3m units cm 6. The perimeter of triangle STU is cm 7. The distance across the river is 51.2 m 8. Side D is 9.6 cm