1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Inessa [10]
1 year ago
11

Applying the the properties of Congruence.

Mathematics
1 answer:
kolbaska11 [484]1 year ago
6 0

Answer:

LM = 35cm and m \angle H = 98^o

Step-by-step explanation:

For this problem, it is critical to understand what the opening statement means.

Quadrilateral GHJK \cong quadrilateral LMNP means that each pair of <u>corresponding sides</u> between shapes are congruent (they have equal length), and each pair of <u>corresponding angles</u> between shapes are congruent (they have equal measure -- or same number of degrees when measured with a protractor).

So, it's important to be able to determine which pairs are the <u>corresponding</u> parts.  When the congruence is given like Quadrilateral GHJK \cong quadrilateral LMNP  the letters are in a particular order.  That order explains the order in which the shape is drawn out, and they put them in the same order for both shapes to mean that those are the letters that correspond with each other.

So, the first letter of each are a corresponding pair, the second letter of each are a corresponding pair, ... etc

In other words, point G and L correspond, point H and K correspond, etc.

<h3><u>Part 1 -- Find LM</u></h3>

To find the corresponding sides, the sides for these shapes are defined by a two adjacent letters (two letters next to each other in the name, or ... as it wraps around the end of the name... the first and last letter).

\overline{GH} \cong \overline{LM}\\ \overline{HJ} \cong \overline{MN} \\\overline{JK} \cong \overline{NP}\\\overline{KG} \cong \overline{PL}

While all of that is true, we only have a few sides with any information in them.  Notice that side GH in the first shape, and sides PL & LM in the second shape are the only sides with anything written.  So, we just need to determine which sides correspond (and thus, are congruent), so that we know how to move forward.  Looking at the list of congruent sides above, we see that  \overline{GH} \cong \overline{LM}

Since the sides (line segments) are congruent, their lengths are equal

GH = LM

...so we can substitute the expressions for each angle into the equation and solve for the unknown value "x"

GH = LM\\(4x+3)=(6x-13)

(4x+3)+13=(6x-13)+13\\4x+16=6x

(4x+16)-4x=(6x)-4x\\16=2x

divide both sides by 2

\frac{16}{2}=\frac{2x}{2}\\8=x

So, to find the length, LM, we just need to look back at the expression for LM:

LM = 6x-13\\LM = 6(8)-13\\LM = 48-13\\LM = 35

remembering that the lengths are measured in centimeters (as indicated on the diagram): LM = 35cm

<h3><u>Part 2 -- Find m∠H</u></h3>

To find the corresponding angles, the angles for these shapes are defined by a single letter, so since the points correspond in order, the names of the shapes tell which angles are congruent.

\angle G \cong \angle L\\\angle H \cong \angle M\\\angle J \cong \angle N\\\angle K \cong \angle P

While all of that is true, we only have a few angles with any information in them.  Notice that ∠H in the first shape, and ∠L & ∠M in the second shape are the only angles with anything written.  So, we just need to determine which angles correspond (and thus, are congruent), so that we know how to move forward.  Looking at the list of congruent angles above, we see that  

\angle H \cong \angle M

Since the angles are congruent, their measures are equal

m\angle H = m\angle M

...so we can substitute the expressions for each angle into the equation and solve for the unknown value "y"

m\angle H = m\angle M\\(9y+17)=(11y-1)

subtract 1 from both sides

(9y+17)+1=(11y-1)+1\\9y+18=11y

subtract 9y from both sides

\\(9y+18)-9y=(11y)-9y\\18=2y

divide both sides by 2

\frac{18}{2}=\frac{2y}{2}\\9=y

So, to find angle H, we just need to look back at the expression for the measure of angle H:

m\angle H = 9y+17\\m\angle H = 9(9)+17\\m\angle H = 81+17\\m\angle H = 98

remembering that the angle is measured in degrees (as indicated on the diagram): m \angle H = 98^o

You might be interested in
- slope = 2/7 ; y-intercept = 0​
Yuri [45]

Answer:

y=2/7x

Step-by-step explanation:

y=mx+b where m=slope and b=y-intercept,

y=2/7x+0

y=2/7x

4 0
3 years ago
7,000 is 100 times what number ?
anzhelika [568]

Answer:

70

Step-by-step explanation:

5 0
3 years ago
Okay someone help me please what is ten times negative three ? ):
mr_godi [17]

Answer:

-30

Step-by-step explanation:

10 x 3 = 30

10 x -3= -30

please mark me brainliest

two negatives makes a positive

two positives markets a negative

and

a positive and a negative makes a negative

5 0
3 years ago
Read 2 more answers
Mona has 400 yards of fencing to enclose a rectangular area. Find the dimensions of the rectangle that maximize the enclosed are
olganol [36]

Answer:

10,000 yd

Step-by-step explanation:

8 0
3 years ago
Given the box plot, will the mean or the median provide a better description of the center? (1 point)
Lady_Fox [76]
The data are bunched towards the higher values above the mean, but spread out to values below the mean. Therefore the median provides a better description of the center.
7 0
2 years ago
Read 2 more answers
Other questions:
  • What number must you add to complete the square?
    7·2 answers
  • I just need the fill in values part
    6·1 answer
  • Mark wrote the highest score he made on his new video game as the product of 70*6000. Use the associative and commutative proper
    11·1 answer
  • What is the value of x in the equation 1.8-3.7x=4.2x+0.3
    6·1 answer
  • Two sides of a triangle measure 8 cm and 15 cm. Which could be the length of the third side?
    14·2 answers
  • Order Of operation.
    6·2 answers
  • Which of the following is a solution to the linear function y = 4/5x+7
    11·1 answer
  • How do we prove that a line is a tangent of a circle ​
    10·1 answer
  • Estimate the quotients 123.8 ÷ 62​
    9·1 answer
  • I need help with this pls
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!