triangles

Triangles : 

Triangles Made By:- Rishabh Gautam Class :- Xth-B Submitted To :- Mrs. Sunita Single

Triangles : 

Triangles A triangle is a 3-sided polygon. Every triangle has three sides and three angles. When added together, the three angles equal 180°.

Triangles : 

Triangles Can be classified by the number of congruent sides

Scalene Triangle : 

Scalene Triangle Has no congruent sides

Isosceles Triangle : 

Isosceles Triangle Has at least two congruent sides

Equilateral Triangle : 

Equilateral Triangle Has three congruent sides

Triangles : 

Triangles Can be classified by the angle measures

Right Triangle : 

Right Triangle Has one right angle

Acute Triangle : 

Acute Triangle Has three acute angles

Obtuse Triangle : 

Obtuse Triangle Triangle with one obtuse angle

Equiangular : 

Equiangular All angles are congruent

Slide 12: 

The Pythagorean Lived in southern Italy during the sixth century B.C. Considered the first true mathematician Used mathematics as a means to understand the natural world First to teach that the earth was a sphere that revolves around the sun

Right Triangles : 

Right Triangles Longest side is the hypotenuse, side c (opposite the 90o angle) The other two sides are the legs, sides a and b Pythagoras developed a formula for finding the length of the sides of any right triangle

The Pythagorean Theorem : 

The Pythagorean Theorem “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” a2 + b2 = c2

Slide 15: 

What are similar triangles?

Slide 16: 

Similar triangles are the same shape but not the same size. This means that corresponding angles of the two triangles are congruent, and that the corresponding sides are in the same ratio. This ratio is called the scale factor.

Slide 17: 

Similar triangles have the following properties: They have the same shape but not the same size. Each corresponding pair of angles is equal. The ratio of any pair of corresponding sides is the same.

Slide 18: 

Similar triangles are triangles that have the same shape but not necessarily the same size. ABC  DEF When we say that triangles are similar there are several repercussions that come from it. A  D B  E C  F

Slide 19: 

1. PPP Similarity Theorem  3 pairs of proportional sides Six of those statements are true as a result of the similarity of the two triangles. However, if we need to prove that a pair of triangles are similar how many of those statements do we need? Because we are working with triangles and the measure of the angles and sides are dependent on each other. We do not need all six. There are three special combinations that we can use to prove similarity of triangles. 2. PAP Similarity Theorem  2 pairs of proportional sides and congruent angles between them 3. AA Similarity Theorem  2 pairs of congruent angles

Slide 20: 

1. PPP Similarity Theorem  3 pairs of proportional sides ABC  DFE

Slide 21: 

2. PAP Similarity Theorem  2 pairs of proportional sides and congruent angles between them mH = mK GHI  LKJ

Slide 22: 

The PAP Similarity Theorem does not work unless the congruent angles fall between the proportional sides. For example, if we have the situation that is shown in the diagram below, we cannot state that the triangles are similar. We do not have the information that we need. Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.

Slide 23: 

3. AA Similarity Theorem  2 pairs of congruent angles mN = mR mO = mP MNO  QRP

Slide 24: 

It is possible for two triangles to be similar when they have 2 pairs of angles given but only one of those given pairs are congruent. mT = mX mS = 180- (34 + 87) mS = 180- 121 mS = 59 mS = mZ TSU  XZY

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THE END THANK YOU