Pssa.20090401.triangle, Exterior Angles.solutions

HOPE Charter School 2009 PSSA Boot Camp

Wednesday, April 1, 2009

PSSA Math Reporting Categories: A Numbers and Operations B Measurement √ C Geometry √ D Algebraic Concepts E Data Analysis and Probability SOLUTIONS Name (legibly printed): _______________________________________________ Seminar (circle one): Ackerman Beals Capers Harvey Hickson Instructions: • Place your solution into box in front of main office by end of 4th period today. • Winner will be announced during 6th period lunch today. Winner will be randomly selected from all entries, but the answer must be correct and justified (SHOW WORK OR EXPLAIN). • Link with questions and solutions posted at http://hopecharter.blogspot.com

Using the following figure, what is the value of ( x + y + z ) − ( a + b + c ) ?

A B C D

-180 0 180 360

Solutions:

Method 1: seeing there are exterior angles in the illustration From geometry, the sum of the three interior angles of a triangle equal 180° , so: a + b + c = 180° [1] This is the right term of the desired binomial expression ( x + y + z ) − ( a + b + c )

From geometry, the exterior angle is the sum of the two opposite interior angles. There are three of these relationships in the above illustration:

x=

b +c

[2]

+c

[3]

y= a z = a +b

[4]

The addition of equations [2], [3], and [4] will yield the left term of the desired binomial expression ( x + y + z ) − ( a + b + c ) , so x + y + z = 2a + 2b + 2c

[5]

x + y + z = 2(a + b + c)

[6]

Substituting [1] and [6] into the original binomial expression yields: ( x + y + z ) − (a + b + c) 2(a + b + c) − (a + b + c) a+b+c 180°

Method 2: not seeing there are exterior angles in the illustration

From geometry, the sum of the three interior angles of a triangle equal 180° , so: a + b + c = 180° [11] This is the right term of the desired binomial expression ( x + y + z ) − ( a + b + c ) From geometry, there are three supplementary relationships in the above illustration: x = 180° − a [12] y = 180° − b z = 180° − c

[13] [14]

The addition of equations [12], [13], and [14] will yield the left term of the desired binomial expression ( x + y + z ) − ( a + b + c ) , so x + y + z = 540° − ( a + b + c )

[15]

Substituting [11] and [15] into the original binomial expression yields: ( x + y + z ) − (a + b + c) ⎡⎣540° − ( a + b + c ) ⎤⎦ − ( a + b + c ) [540° − 180°] − 180° 180°