Download Similar triangles Two polygons are similar if corresponding angles

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Similar triangles Two polygons are similar if corresponding angles are congruent and corresponding sides are proportional. For triangles we have a postulate and two theorems we can use to show that triangles are similar. Angle-­‐Angle Similarity Postulate (AA~) If two pairs of corresponding angles are congruent, then triangles are similar. Example: B W 26o 64o Y o o 2
6
6
4
M F D Since ∠M ≅ ∠F and ∠D ≅ ∠Y , then ΔMBD ~ ΔFWY because of AA~. Side-­‐Angle-­‐Side Similarity Theorem (SAS~) If two pairs of corresponding sides are proportional and the included angles are congruent, then triangles are similar. Example: ∠NRK ≅ ∠GRP because of vertical angles theorem. NR 9 3
KR 15 3
= = and = = , then sides are proportional. GR 6 2
PR 10 2
Therefore ΔNKR ~ ΔGPR because of SAS~. Side-­‐Side-­‐Side Similarity Theorem (SSS~) If three pairs of corresponding sides are proportional, then triangles are similar. Example: AB 9 3
AC 6 3
BC 6 3
= = and
= = and
= = , the sides are proportional. EF 12 4
EG 8 4
FG 8 4
Therefore ΔABC ~ ΔEFG because of SSS~. Examples. Are the triangles similar? If so, write a similarity statement and name the postulate or theorem used. #1 The triangles are not similar since ∠X is not congruent to ∠R . #2 AB 8 2
AE 6 2
= = and
= = so sides are ∠AEB ≅ ∠CED because of vertical angles theorem. CD 12 3
CE 9 3
proportional. But the triangles are not similar because the angle is not between the two sides. #3 ∠FHG ≅ ∠KHJ because of vertical angles theorem. ∠F ≅ ∠K because of alternate interior angles theorem. ΔFHG ~ ΔKHJ because of AA~. #4 MO 6 2
OR 10 2
= = and
= = , so sides are ∠O ≅ ∠H because all right angles are congruent. GH 3 1
HI 5 1
proportional. ΔMOR ~ ΔGHI #5 When you are working with overlapping figures like the one above, redraw the two separate triangles. AX 20 4
AY 25 5
Because the figures are overlapping, ∠A ≅ ∠A . =
= and
=
= , the sides are not AB 45 9
AC 55 11
proportional . The triangles are not similar. #6 Find the value of the variable. Since the triangles are similar because of AA~, then the corresponding sides are proportional. 5 3
=
x 4
3x = 20 20
x=
3
#7 Find the value of the variable. Since the triangles are similar by AA~, then the corresponding sides are proportional. 22 24
=
x 14
24x = 308 77
x=
6
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