Question

fluency

1. if ( cd = fg ) and ( de = gh ), then which of the following would justify that ( cd + de = fg + gh )?

(1) the partition property
(2) the addition property
(3) the subtraction property
(4) the substitution property

1. in the diagram below, ( overrightarrow{rs} ), ( overrightarrow{rt} ), and ( overrightarrow{ru} ) are all drawn from vertex ( r ). from this we can state that ( mangle sru = mangle srt + mangle tru ). this is an example of which of the following?

(1) the partition property
(2) the addition property
(3) the subtraction property
(4) the substitution property
(diagram: angle with vertex r, sides rs, rt, ru)

1. in a proof, it is known that ( overline{abcd} ) and ( ab + bc = bc + cd ). which of the following would allow us to conclude that ( overline{ab} ) and ( overline{cd} ) have the same lengths?

(1) the partition property
(2) the addition property
(3) the subtraction property
(4) the substitution property

1. in the following diagram, point ( b ) lies on ( overline{ac} ) such that ( overline{bd} perp overline{ac} ) as shown. from this we can conclude that ( mangle dba = mangle dbc ). the reason will be: all right angles are of equal measure (or are congruent). this is simply a special case of which of the following:

(1) the partition property
(2) the addition property
(3) the subtraction property
(4) the substitution property
(diagram: triangle with d at top, b on ac, right angles at b)

1. which of the following is not used to prove vertical angles have equal measures?

(1) the partition property
(2) the addition property
(3) the subtraction property
(4) the substitution property

n-gen math® geometry – unit 3 – euclidean triangle geometry – lesson 7 (v.10.14.24)
emathinstruction, red hook, ny 12571, © 2023

Explanation:

Question 1

Step1: Recall Properties

The Substitution Property states that if \( a = b \), then \( a \) can be substituted for \( b \) (or vice versa) in any equation. Here, \( CD = FG \) and \( DE = GH \), so we substitute \( FG \) for \( CD \) and \( GH \) for \( DE \) in \( CD + DE \) to get \( FG + GH \). The Addition Property is for adding equal quantities, Partition is about parts of a whole, Subtraction is for subtracting equal quantities. So Substitution applies here.

Step2: Identify Correct Option

From the analysis, the Substitution Property (option 4) is the correct justification.

Step1: Recall Properties

The Partition Property (Angle Addition Postulate is a type of Partition Property) states that if a ray is inside an angle, the measure of the larger angle is the sum of the measures of the two smaller angles formed. Here, \( \overrightarrow{RT} \) is inside \( \angle SRU \), so \( m\angle SRU = m\angle SRT + m\angle TRU \) follows the Partition Property. Addition is for equal quantities, Subtraction for subtracting, Substitution for replacing values.

Step2: Identify Correct Option

The Partition Property (option 1) is the correct example.

Step1: Recall Properties

The Subtraction Property of Equality states that if \( a + b = b + c \), then subtracting \( b \) from both sides gives \( a = c \). Here, we have \( AB + BC = BC + CD \), so subtracting \( BC \) from both sides (Subtraction Property) gives \( AB = CD \). Partition is about parts, Addition for adding, Substitution for replacing.

Step2: Identify Correct Option

The Subtraction Property (option 3) allows us to conclude \( AB = CD \).

Answer:

(4) The Substitution Property

Question 2