Unit 3 Triangles

Classifying Triangles Lesson 3.1 Classifying Triangles

Lesson 3.1 Objectives Classify triangles according to their side lengths. (G1.2.1) Classify triangles according to their angle measures. (G1.2.1) Find a missing angle using the Triangle Sum Theorem. (G1.2.2) Find a missing angle using the Exterior Angle Theorem. (G1.2.2)

Classification of Triangles by Sides Equilateral Isosceles Scalene Looks Like Characteristics 3 congruent sides 2 congruent sides No Congruent Sides

Classification of Triangles by Angles Name Acute Equiangular Right Obtuse Looks Like Characteristics ALL acute angles ALL congruent angles ONLY 1 right angles ONLY 1 obtuse angle

Example 3.1 Classify the following triangles by their sides and their angles. Scalene Obtuse Scalene Right Isosceles Acute Equilateral Equiangular

Vertex The vertex of a triangle is any point at which two sides are joined. It is a corner of a triangle. There are 3 in every triangle

How to Name a Triangle To name a triangle, simply draw a small triangle followed by its vertices. We usually try to name the vertices in alphabetical order, when possible. Example: ABC

More Parts of Triangles If you were to extend the sides you will see that more angles would be formed. So we need to keep them separate There are three angles called interior angles because they are inside the triangle. There are three new angles called exterior angles because they lie outside the triangle.

Theorem 4.1: Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180o. mA + mB + mC = 180o

Example 3.2 Solve for x and then classify the triangle based on its angles. Acute 75o 50o 3x + 2x + 55 = 180 Triangle Sum Theorem 5x + 55 = 180 Simplify 5x = 125 SPOE x = 25 DPOE

Example 3.3 Solve for x and classify each triangle by angle measure. Right Acute

Example 3.4 Draw a sketch of the triangle described. Mark the triangle with symbols to indicate the necessary information. Acute Isosceles Equilateral Right Scalene

Example 3.5 Draw a sketch of the triangle described. Mark the triangle with specific angle measures, side lengths, or symbols to indicate the necessary information. Obtuse Scalene Right Isosceles Right Equilateral (Not Possible)

Theorem 4.2: Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Example 3.6 Solve for x Exterior Angles Theorem Combine Like Terms Subtraction Property Addition Property Division Property

Corollary to the Triangle Sum Theorem A corollary to a theorem is a statement that can be proved easily using the original theorem itself. This is treated just like a theorem or a postulate in proofs. The acute angles in a right triangle are complementary.

Example 3.7 Find the unknown angle measures. VA VA If you don’t like the Exterior Angle Theorem, then find m2 first using the Linear Pair Postulate. Then find m1 using the Angle Sum Theorem. VA VA

Homework 3.1 Lesson 3.1 – All Sections p1-6 Due Tomorrow

Inequalities in One Triangle Lesson 3.2 Inequalities in One Triangle

Lesson 3.2 Objectives Order the angles in a triangle from smallest to largest based on given side lengths. (G1.2.2) Order the side lengths of a triangle from smallest to largest based on given angle measures. (G1.2.2)

Theorem 5.10: Side Lengths of a Triangle Theorem If two sides of a triangle unequal, then the measures of the angles opposite theses sides are also unequal, with the greater angle being opposite the greater side. Basically, the largest angle is found opposite the largest side. Basically, the largest side is found opposite the largest angle. 2nd Largest Angle Longest side Smallest Side Smallest Angle Largest Angle 2nd Longest Side

Theorem 5.11: Angle Measures of a Triangle Theorem If two angles of a triangle unequal, then the measures of the sides opposite theses angles are also unequal, with the greater side being opposite the greater angle. Basically, the largest angle is found opposite the largest side. Basically, the largest side is found opposite the largest angle. 2nd Largest Angle Longest side Smallest Side Smallest Angle Largest Angle 2nd Longest Side

Example 3.8 Order the angles from largest to smallest.

Example 3.9 Order the sides from largest to smallest. 33o

Example 3.10 Order the angles from largest to smallest. In ABC AB = 12 BC = 11 AC = 5.8 Order the sides from largest to smallest. In XYZ mX = 25o mY = 33o mZ = 122o

Homework 3.2 Lesson 3.2 – Inequalities in One Triangle Due Tomorrow p7-8 Due Tomorrow Quiz Friday, October 15th

Isosceles and Equilateral Triangles Lesson 3.3 Isosceles and Equilateral Triangles

Lesson 3.3 Objectives Utilize the Base Angles Theorem to solve for angle measures. (G1.2.2) Utilize the Converse of the Base Angles Theorem to solve for side lengths. (G1.2.2) Identify properties of equilateral triangles to solve for side lengths and angle measures. (G1.2.2)

Special Parts of an Isosceles Triangle An isosceles triangle has only two congruent sides Those two congruent sides are called legs. The third side is called the base. legs base

Isosceles Triangle Theorems Theorem 4.6: Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent to each other. Theorem 4.7: Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent.

Example 3.11 Solve for x and y. + = 90o = 90o = 45 45 = 45o 55o 55o

Equilateral Triangles Corollary to Theorem 4.6 If a triangle is equilateral, then it is equiangular. Corollary to Theorem 4.7 If a triangle is equiangular, then it is equilateral.

Example 3.12 Solve for x and y. 5xo 5xo It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest! Or…In order for a triangle to be equiangular, all angles must equal…

Homework 3.3 Lesson 3.3 – Isosceles and Equilateral Triangles p9-11 Due Tomorrow Quiz Tomorrow Tuesday, October 19th

Medians And Altitudes of Triangles Lesson 3.4 Medians And Altitudes of Triangles

Lesson 3.4 Objectives Identify a median, an altitude, and a perpendicular bisector of a triangle. (G1.2.5) Identify a centroid of a triangle. Utilize medians and altitudes to solve for missing parts of a triangle. (G1.2.5) Identify the orthocenter of a triangle.

Perpendicular Bisector A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector.

Triangle Medians A median of a triangle is a segment that does the following: Contains one endpoint at a vertex of the triangle, and Contains its other endpoint at the midpoint of the opposite side of the triangle. A B C D

Centroid Remember: All medians intersect the midpoint of the opposite side. When all three medians are drawn in, they intersect to form the centroid of a triangle. This forms a point of concurrency which is defined as a point formed by the intersection of two or more lines. The centroid happens to find the balance point for any triangle. In Physics, this is how we locate the center of mass. Obtuse Acute Right

Theorem 5.7: Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. The centroid is 2/3 the distance from any vertex to the opposite side. Or said another way, the centroid is twice as far away from the opposite angle as it is to the nearest side. AP = 2/3AE BP = 2/3BF CP = 2/3CD

Example 3.13 S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the following: RV 6 SU 2 Half of 4 is 2 RU 4 + 2 = 6 RW 12 TS 6 is 2/3 of 9 SV 3 Half of 6, which is the other part of the median.

Altitudes An altitude of a triangle is the perpendicular segment from a vertex to the opposite side. It does not bisect the angle. It does not bisect the side. The altitude is often thought of as the height. While true, there are 3 altitudes in every triangle but only 1 height!

Orthocenter The three altitudes of a triangle intersect at a point that we call the orthocenter of the triangle. The orthocenter can be located: inside the triangle outside the triangle, or on one side of the triangle Obtuse Right Acute The orthocenter of a right triangle will always be located at the vertex that forms the right angle.

Example 3.14 Is segment BD a median, altitude, or perpendicular bisector of ABC? Hint: It could be more than one! Perpendicular Bisector Altitude Median Median None None

Homework 3.4 Lesson 3.4 – Altitudes and Medians p12-13 Due Tomorrow

Area and Perimeter of Triangles Lesson 3.5 Area and Perimeter of Triangles

Lesson 3.5 Objectives Find the perimeter and area of triangles. (G1.2.2)

Reviewing Altitudes Determine the size of the altitudes of the following triangles. 6 16 If it is a right triangle, then you can use Pythagorean Theorem to solve for the missing side length. ?

Area The area of a figure is defined as “the amount of space inside the boundary of a flat (2-dimensional) object” http://www.mathsisfun.com/definitions/area.html Because of the 2-dimensional nature, the units to measure area will always be “squared.” For example: in2 or square inches ft2 or square feet m2 or square meters mi2 or square miles The area of a rectangle has up until now been found by taking: length x width (l x w) We will now change the wording slightly to fit a more general pattern for all shapes, and that is: base x height (b x h) That general pattern will exist as long as the base and height form a right angle. Or said another way, the base and height both touch the right angle. l b h w

Area of a Triangle The area of a triangle is found by taking one-half the base times the height of the triangle. Again the base and height form a right angle. Notice that the base is an actual side of the triangle, and… The height is nothing more than the altitude of the triangle drawn from the base to the opposite vertex. b h h b

Perimeter of a Triangle The perimeter of a triangle is found by taking the sum of all three sides of the triangle. So basically you need to add all three sides together. The perimeter is a 1-dimensional measurement, so the units should not have an exponent on them. Example: in ft m mi b a h c

Example 3.15 Find the area and perimeter of the following triangles.

Homework 3.5 Lesson 3.5 – Area and Perimeter of Triangles Due Tomorrow