Similar figures have the same shape but may have different sizes. Two triangles are similar if their corresponding angles are congruent and the lengths of their corresponding sides are proportional.

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

**Step 1 : **

Use your protractor and a straightedge to draw a triangle. Make one angle measure 45° and another angle measure 60°.

**Step 2 : **

Compare your triangle to those drawn by your classmates. Triangle drawn by you and the triangles drawn by your classmates will have same shape and different in sizes.

**Step 3 : **

Use the Triangle Sum Theorem to find the measure of the third angle of your triangle.

180° - (45° + 60°) = 180° - 105° = 75°

1. If two angles in one triangle are congruent to two angles in another triangle, what do you know about the third pair of angles ?

They must also be congruent.

2. Are two pairs of congruent angles enough information to conclude that two triangles are similar ? Explain.

Yes

By the Triangle Sum Theorem, the third pair of angles must have the same angle measure and thus are congruent, so the triangles must be similar.

**Problem 1 : **

Explain whether the triangles PQR and STU are similar.

**Solution :**

The figure shows only one pair of congruent angles. Find the measure of the third angle in each triangle.

**Triangle PQR : **

Write the Triangle Sum Theorem for this triangle.

m∠P + m∠Q + m∠R = 180°

Substitute the given angle measures.

45° + 100° + m∠R = 180°

Simplify.

145° + m∠R = 180°

Subtract 145° from both sides.

145° + m∠R - 145° = 180° - 145°

Simplify.

m∠R = 35°

**Triangle STU : **

Write the Triangle Sum Theorem for this triangle.

m∠S + m∠T + m∠U = 180°

Substitute the given angle measures.

m∠S + 100° + 35° = 180°

Simplify.

m∠S + 135° = 180°

Subtract 135° from both sides.

m∠S + 135° - 135° = 180° - 135°

Simplify.

m∠S = 45°

**Conclusion : **

Three Angles of triangle PQR are 45°, 100° and 35°.

Three Angles of triangle STU are 45°, 100° and 35°.

Because two angles in one triangle are congruent to two angles in the other triangle, the two triangles are similar.

**Problem 2 : **

Explain whether the triangles ABC and DEF are similar.

**Solution :**

The figure shows only one pair of congruent angles. Find the measure of the third angle in each triangle.

**Triangle ABC : **

Write the Triangle Sum Theorem for this triangle.

m∠A + m∠B + m∠C = 180°

Substitute the given angle measures.

m∠A + 58° + 70° = 180°

Simplify.

m∠A + 128° = 180°

Subtract 128° from both sides.

m∠A + 128° - 128° = 180° - 128°

Simplify.

m∠A = 52°

**Triangle DEF : **

Write the Triangle Sum Theorem for this triangle.

m∠D + m∠E + m∠F = 180°

Substitute the given angle measures.

70° + m∠E + 49° = 180°

Simplify.

m∠E + 119° = 180°

Subtract 119° from both sides.

m∠E + 119° - 119° = 180° - 119°

Simplify.

m∠E = 61°

**Conclusion :**

Three Angles of triangle ABC are 52°, 58° and 70°.

Three Angles of triangle DEF are 70°, 61° and 49°.

Because only one angle is congruent, the two triangles are not similar.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**