GEOMETRYTopic : Angles and TrianglesReporter :Aida A. MaribbayProfessor:Dr. Meriam Dela Cruz

THE BASIC TERMSANGLES An angle is the union of tworays which have the same endpoint The two rays are called the sides of the angle; the common endpoint is the vertex.

AB and BC are sides of

An angle separates a plane intothree sets** the points on the angle** those in the interior of the angle** those in the exterior of the angle

The interior of an angle is the intersection of two half planes The exterior of an angle is the set of points in a plane whichdo not belong to the interior ofthe angle or to the angle itselfinteriorexteriorexterior

WAYS IN NAMING AN ANGLE* Three lettersBy using* Numbers* Greek Letters* Radian* Angle Measusrement* Vertex

Measuring An Angle: An Angle is measured with a Protractor: The unit measure is called a Degree:The no. of degrees in an angle is called its Measure

Kinds of AnglesA. Acute angleMeasures greater than 0 but less than 90

B. Right AngleIs an angle whose measure is90

C. Obtuse Angle Measures greater than 90 but less than 180

D. Straight Angle Measures exactly 180

E. Reflex angle Is an angle whose measure isbetween 180 to 360. one complete revolution measures exactly 360

Angle PairsComplementary AnglesTwo angles are complimentary if and only if the sum of their measure is 90

Supplementary Angles Two angles are suplementary if and if the sum of their measures is 180

Vertical AnglesAre angles whose sides form twopairs of opposite raysFor any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles.

Vertical angles have the same degree measurement. Angle BEC and angle AED are also vertical angles.

Alternate Exterior Angles

For any pair of parallel lines 1and 2, that are both intersectedby a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles.

Alternate exterior angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.

Alternate Interior Angles

For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles.

Alternate interior angles have the same degree measurement. Angle B and angle C are also alternate interior angles.

Corresponding Angles

For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles.

Corresponding angles have the same degree measurement. Angle B and angle D are also corresponding angles.

Angle Bisector

An angle bisector is a ray that divides an angle into two equal angles. The blue ray on the right is the angle bisector of the angle on the left.

Perpendicular Lines

Two lines that meet at a right angle are perpendicular

Linear Pair of AnglesA pair of adjacent angles formed by intersecting lines. Angles 1 and 2 below are a linear pair. So are angles 2 and 4, angles 3 and 4, and angles 1 and 3. Linear pairs of angles are supplementary.

Congruent Angles

Congruent Angles have the same angle in degrees.

TRIANGLESA Triangle is the union of three segments determined by three noncollinear pointsThe non collinear points are calledVertices, the three segments are theSides of the triangle. Every triangle determines three angles calledAngles of the triangles

ABCThe sides of triangle ABC are line ofAB,AC, and BC. The three angles are angle A, angle B. Angle C.

A triangle separates the plane where it is drawn into three sets:The triangle:The interior The interior of a triangle is the intersection of the interior of itsangle

:The exterior The exterior is the set of points in a plane which do not belong exteriorinterior

Types of Triangles

Classification of triangle in terms of anglesRight Triangles A right triangle has one 90 90

Acute Triangle TheAcute Triangle has three acute angle .

Obtuse Triangle The Obtuse Triangle has an obtuse angle

606060Equiangular TriangleAll three angles are equal to 60 degrees

Equilateral Triangle: All three sides have equal length Classification of triangle in terms of sides

Scalene Triangle The Scalene Triangle has no congruent sides.

Isosceles triangle The isosceles triangle has twoCongruent sides

Secondary Part of a TriangleAngle Bisectors in a Triangle

The angle bisector of an angle of a triangle is a straight line that divides the angle into two congruent angles.

Perpendicular bisector

The perpendicular bisector of a side of a triangle is a line perpendicular to the side and passing through its midpoint. A

Find two angles such that The angles are supplementary and the larger is twice the smaller.b. The angles are complementary and the larger is 20 more than the smaller.

The angles are adjacent and forms an angle of 120. The larger is 20 less than three times the smaller.The angles are vertical and complementary.

(a)(b)(c)(d)

Solutions:In each solution, x is a number only. This number indicates the number of degrees contained in the angle. Hence, if x=60, the angle measures 60.

Let x=m (smaller angle) and 2x=m (larger angle)Principle 5: x + 2x= 180, so 3x=180; x=602x=120 Ans: 60 and 120b)Let x=m (smaller angle) and x + 20=m (larger angle)Principle 3: x + (x+20)= 90, or 2x + 20=90; x=35x+20=55 Ans: 35 + 55Let x=m (smaller angle) and 3x-20=m (larger angle)Principle 3: x + (3x-20)= 120, or 4x 20=120; x=35 $ 3x-20=85 Ans: 35 + 85Let x=m (each vertical angle) . They are congruent by principle 2Principle 3: x + x= 90, or 2x=90; x=45 Ans: 35 + 55