Unit 3 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)
Example 3.1 Classify the following triangles by their sidesand 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 interiorangles of a triangle is 180o. mA + mB + mC = 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 twononadjacentinterior 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 If you don’t like the Exterior Angle Theorem, then find m2 first using the Linear Pair Postulate. Find the unknown angle measures. Then find m1 using the Angle Sum Theorem. VA VA
Homework 3.1 • Lesson 3.1 – All Sections • p1-6 • Due Tomorrow
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 anglesopposite theses sides are also unequal, with the greater angle being opposite the greater side. • Basically, the largestangle is found opposite the largestside. • Basically, the largestside is found opposite the largestangle. 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 sidesopposite theses angles are also unequal, with the greater side being opposite the greater angle. • Basically, the largestangle is found opposite the largestside. • Basically, the largestside is found opposite the largestangle. 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 ABCAB = 12BC = 11AC = 5.8 • Order the sides from largest to smallest. • In XYZmX = 25omY = 33omZ = 122o
Homework 3.2 • Lesson 3.2 – Inequalities in One Triangle • p7-8 • Due Tomorrow • Quiz Friday, October 15th
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 anglesopposite them are congruent to each other. Theorem 4.7: Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sidesopposite them are congruent.
Example 3.11 + 45o 45o = 90o Solve for x and y. 55o 55o = 90o = 45 45 = 75o
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
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 B C • 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. D
Centroid Remember: All medians intersect the midpointof 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 thecenter 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 • 6 • 4 + 2 = 6 • RW • 12 • TS • 6 • 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 perpendicularsegment from a vertex to the opposite side. • It does not bisectthe angle. • It does not bisect the side. • The altitude is often thought of as the height. • While true, there are 3 altitudesin every triangle but only 1 height!
Orthocenter • The three altitudesof 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 Median Altitude Median None None
Homework 3.4 • Lesson 3.4 – Altitudes and Medians • p12-13 • Due Tomorrow
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 areaof 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: • in2or square inches • ft2or square feet • m2or square meters • mi2or 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 asthe baseand heightform a right angle. • Or said another way, the base andheight both touch the right angle. l b h w