TheoremIf two sides of a triangle are not congruent, thenthe larger angle is opposite the longer side.TheoremIf two angles of a triangle are not congruent, thenthe longer side is opposite the larger angle.TriangleInequalityTheoremThe sum of any two side lengths of a triangle isgreater than the third side length.HingeTheoremIf two sides of one triangle are congruent to twosides of another triangle and the third sides are notcongruent, then the longer third side is across fromthe larger included angle.Converse ofHingeTheoremIf two sides of one triangle are congruent to twosides of another triangle and the third sides are notcongruent, then the larger included angle is acrossfrom the longer third side.Converse ofthePythagoreanTheoremIf the sum of the squares of the lengths of twosides of a triangle is equal to the square of thelength of the third side, then the triangle is a righttriangle.PythagoreanInequalitiesTheoremIn∆ABC, c is the length of the longest side. If c² >a² + b², then∆ABC is an obtuse triangle. If c² < a²+ b², then∆ABC is acute.45˚-45˚-90˚TriangleTheoremIn a 45˚-45˚-90˚triangle, both legs are congruent,and the length of the hypotenuse is the length of alength times the square root of 2.30˚-60˚-90˚TriangleTheoremIn a 30˚-60˚-90˚triangle, the length of thehypotenuse is 2 times the length of the shorter leg,and the length of the longer leg is the length of theshorter leg times the square root of 3.Law of SinesFor any triangle ABC with side lengths a, b, and c,cCbBaAsinsinsin==Law ofCosinesFor any triangle, ABC with sides a, b, and c,CacbacBaccabAbccbacos2,cos2,cos2222222222−+=−+=−+= Show The following costs relate to Richard Industries for the last quarter:Conversion cost. P435,000Direct materials 215,600Factory overhea … d is applied at 60% of direct labor cost. .Selling and administrative expense 185,000What is Richard's prime cost for last quarter? Show solution.A. 460,000B. 410,000C. 405,000D. 487,475 You have just seen that if a triangle has equal sides, the angles opposite these sides are equal, and if a triangle has equal angles, the sides opposite these angles are equal. There are two important theorems involving unequal sides and unequal angles in triangles. They are: Theorem 36: If two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal, and the greater angle is opposite the greater side. Theorem 37: If two angles of a triangle are unequal, then the measures of the sides opposite these angles are also unequal, and the longer side is opposite the greater angle. Example 1: Figure 1 shows a triangle with angles of different measures. List the sides of this triangle in order from least to greatest.
 Figure 1 List the sides of this triangle in increasing order. Because 30° < 50° < 100°, then RS < QR < QS. Example 2: Figure 2 shows a triangle with sides of different measures. List the angles of this triangle in order from least to greatest.
Figure 2 List the angles of this triangle in increasing order. Because 6 < 8 < 11, then m ∠ N < m ∠ M < m ∠ P. Example 3: Figure 3 shows right Δ ABC. Which side must be the longest?
Figure 3 Identify the longest side of this right triangle. Because ∠ A + m ∠ B + m ∠ C = 180 ° (by Theorem 25) and m ∠ = 90°, we have m ∠ A + m ∠ C = 90°. Thus, each of m ∠ A and m ∠ C is less than 90°. Thus ∠ B is the angle of greatest measure in the triangle, so its opposite side is the longest. Therefore, the hypotenuse, AC , is the longest side in a right triangle. Which if two angles of a triangle are not congruent then the larger side is?Theorem If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.
When one side of a triangle is longer than the other side the angle opposite the?Theorem: If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.
Is the statement angle opposite to longer side of a triangle is larger True or false?The given statement is true. In a triangle, the length of the side opposite to the larger angle will be greater than the length of side opposite to smaller angle.
What is the larger angle of a triangle?According to the triangle inequality theorem, in a triangle, the largest angle is the one opposite to the longest side of the triangle. Therefore, in triangle XYZ, we have the greatest angle as angle Z.
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