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SD School Geometry Program 3: Triangles QuikNotes The reflexive property states that a value is equal to itself. The substitution property says that we can switch two values that are equal. For example, if the measure of angle 1 plus the measure of angle 3 equals 180, and the measure of angle 1 is equal to the measure of angle 2, the measure of angle 2 plus the measure of angle 3 equals 180. Switching the values of the measure of angle 1 and the measure of angle 2 is substitution. The transitive property says that if A equals B, and B equals C, then A equals C. Theorem: If two angles are the complements of two congruent angles, then the two angles are congruent. For example: If angle 1 and angle 2 are the complements of angle 3 and angle 4, and angle 3 and angle 4 are each 60 degrees, then angle 1 and angle 2 must each be 30 degrees. Theorem: If two angles are the supplements of two congruent angles, then the two angles are congruent. Theorem: every angle has exactly one bisector. As angle CUT is bisected by ray UP, we get two congruent angles, CUP and TUP The parallel postulate is one of the most important postulates in geometry. It says: given a line, L, and a point, P, not on that line, there’s exactly one line, M, through the point, P, that is parallel to line L.” Student Notes SD School Geometry Program 3: Triangles QuikNotes When parallel lines are cut by the transversal, eight different angles are formed. There are four different ways to categorize these angles. The most general way is as corresponding angles. Angles 1 and 5, 2 and 6, 3 and 7, and 4 and 8 are corresponding angles because they are in the same position on either line. They correspond to each other. Angles 3 and 5, and 4 and 6 are alternate interior angles. They’re all interior angles because they’re between the two lines. The pairs we mentioned are alternate interior angles because they are on alternate, or opposite, sides of the transversal. Angles 1 and 7, and 2 and 8 are alternate exterior angles. These guys are all exterior angles because they are on the outside of the lines. Again, the pairs are alternate exterior angles because they are on opposite sides of the transversal.” Angles 3 and 6, and 4 and 5 are same-side interior angles. Because, well, the pairs are all interior angles on the same side of the transversal The Corresponding Angle Postulate: says that lines are parallel if and only if corresponding angles are congruent. Student Notes SD School Geometry Program 3: Triangles QuikNotes A triangle is the figure formed by three segments joining three non-collinear points, which means it’s a three-sided shape. The three segments are called the triangle’s sides. Each side meets another at a vertex, where they form an angle. There are two ways to classify triangles. Either by lengths of their sides, or by their angles. Equilateral triangles have three congruent sides. Like triangle A here. All sides measure 4 inches. A Isosceles triangles have two congruent sides. Triangle B has two sides that equal 4 inches. We call the other side the base. The base can be on any side. For example, if we turn triangle B on its side, or upsidedown, its base is still 2 inches. B C Triangle C is scalene, which means it has no congruent sides. It’s just a mess. It’s all over the place. Equiangular triangles have all three angles congruent. In triangle A, each angle measures 60 degrees. All equiangular angles measure 60 degrees. All equiangular triangles are equilateral, and vice-versa. Student Notes SD School Geometry Program 3: Triangles QuikNotes Right triangles have one right angle. Other parts of right triangle are its two legs, which form the right angle, and the side opposite the right angle is called the hypotenuse. Obtuse triangles have one obtuse angle. This triangle has angles that measure 30 degrees, 35 degrees, and the obtuse angle, which is 115 degrees. Acute triangles have three acute angles. This triangle has measures of 40, 60, and 80 degrees. Student Notes SD School Geometry Program 3: Triangles QuikNotes Complementary angles are two angles whose measures add up to 90 degrees. The measure of angle Y is 33 degrees, and angle O is 57 degrees. If we add the measures of angle Y and angle O together, we get 90 degrees, the measure of a right angle. So angles Y and O are complementary. Supplementary angles are two angles whose measures add up to 180 degrees. The measure of angle D is 99 degrees, and angle A is 81 degrees. We add angle D and angle A together, for a total of 180 degrees, which is the measure of a straight angle. So, angle D and angle A are supplementary Complementary and supplementary angles do NOT need to be adjacent. Vertical angles are what we get when two lines intersect. Vertical angles are two angles that occur opposite each other when two lines intersect. When these two lines intersect, they form four angles. In this diagram, angles 1 and 2 are vertical angles. So are angles 3 and 4. Vertical angles are congruent, if you know the measure of one, you also know the measure of the other. Student Notes SD School Geometry Program 3: Triangles QuikNotes Example: If we’re given the measure of one of these angles, we can figure out the other three. If angle 1 is 55 degrees, then angle 2 is also 55 degrees, because they’re vertical angles. Angles 1 and 3 are supplementary angles. Together, their measures will add up to 180 degrees. We know the measure of angle 1 is 55 degrees, so angle 3 measures 180 degrees minus 55 degrees, or 125 degrees. Angle 4 also measures 125 degrees because angles 3 and 4 are vertical angles An angle bisector is a ray that splits an angle in the middle, forming two congruent adjacent angles. Here, ray QR bisects angle SQT. Angle SQR is congruent to angle TQR. Note the curved symbol inside each of the angles. If there’s a slash in each one, then that means the angles are congruent. Perpendicular lines are intersecting lines that form right angles. Rays and segments can also be perpendicular. Here we have two perpendicular lines, L and M. So all four angles, D, O, G, and S, are right angles. When we write that two lines are perpendicular, we use a special symbol. L is perpendicular to M. l⊥m Student Notes SD School Geometry Program 3: Triangles QuikCheck True or False 1. The transitive property says that if X equals Y, and Y equals Z, then X equals Z. _______ 2. When two lines are cut by a line, we call this intersecting line a bisector. __________ 3. Corresponding angles are in the same position between two parallel lines and their transversal. _____________ 4. Obtuse triangles have three obtuse angles. ____________ 5. Supplementary angles are two angles whose measures add up to 180 degrees. ________ 6. An angle bisector is a ray that splits an angle in the middle, forming two congruent adjacent angles. __________ Q&A 7. If angles J and K are alternate exterior angles, and angle J measures 540 , what is the measure of angle K? 8. If angles M, N, and O are in an acute triangle, and angles M and N are both acute, is angle O acute or obtuse? 9. If angle UVW is 940 , and line TV is UVW’s bisector, what is the measure of angle UVT? 10. Angle 4 is 610 and angles 4 and 5 are vertical angles, what is the measure of angle 5? 11. If an equilateral triangle has angles A, B, and C, and angle A is 600 , what is the measure of angle B? 12. If an isosceles triangle has one angle of 1400 , and one angle 200 , what is the third angle? 13. If angle G is 790 , what is the measure of its supplement, angle H? 14. If a right triangle has one angle 330 and one angle 570 , what is the third angle? SD School Geometry Program 3: Triangles QuikCheck Answer Key True or False 1. The transitive property says that if X equals Y, and Y equals Z, then X equals Z. True 2. When two lines are cut by a line, we call this intersecting line a bisector. False When two lines are cut by a line, we call this intersecting line a transversal. 3. Corresponding angles are in the same position between two parallel lines and their transversal. True 4. Obtuse triangles have three obtuse angles. FalseObtuse triangles have one obtuse angle. 5. Supplementary angles are two angles whose measures add up to 180 degrees. True 6. An angle bisector is a ray that splits an angle in the middle, forming two congruent adjacent angles. True Q&A 7. If angles J and K are alternate exterior angles, and angle J measures 540 , what is the measure of angle K? 54 0 Alternate exterior angles are the same size. 8. If angles M, N, and O are in an acute triangle, and angles M and N are both acute, is angle O acute or obtuse? Acute. All the angles in an acute triangle are acute. 9. If angle UVW is 940 , and line TV is UVW’s bisector, what is the measure of angle UVT? It is one half of 940 . 47 0 10. Angle 4 is 610 and angles 4 and 5 are vertical angles, what is the measure of angle 5? 61 0 Vertical angles have the same measurement. 11. If an equilateral triangle has angles A, B, and C, and angle A is 600 , what is the measure of angle B? 60 0 All the angles of an equilateral triangle are 600 . 12. If an isosceles triangle has one angle of 1400 , and one angle 200 , what is the third angle? 20 0 An isosceles triangle has two angles of the same measurement, and a triangle can only ever have one obtuse angle. 0 13. If angle G is 79 , what is the measure of its supplement, angle H? Supplementary angles add up to 1800 . 81 0 14. If a right triangle has one angle 330 and one angle 570 , what is the third angle? 90 0 All right triangles have one right angle. 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