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]
2 years ago
11

Applying the the properties of Congruence.

Mathematics
1 answer:
kolbaska11 [484]2 years 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
100 points
dusya [7]
A. This expression has 4 terms
B. The constants are -5, 3, 72 and 10
C. The third term is 72
D. The coefficient is y
3 0
2 years ago
How many sides is a decagon? I forgot LOL
umka21 [38]

Answer:

10 sides.

Step-by-step explanation:

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting regular decagon is known as a decagram.

3 0
3 years ago
What is the best first step for solving the given system using substitution while avoiding fractions
Eduardwww [97]

Hi, hope you’re having a good day.

You should probably add an attachment.

Please do! Thanks, Have a great day! ❤️❤️

6 0
3 years ago
Tabitha and her parents drive to Dallas for a wedding. After 90 minutes, Tabithas father tells Her they have traveled 105 miles
scoray [572]
I think the answer is C
8 0
3 years ago
Marcus is playing a carnival game where he must guess the color of a candy bean before he pulls it out of a hat. He gets five tr
BaLLatris [955]
The answer is A 1 point(s)
7 0
3 years ago
Other questions:
  • Problem number 26 of the Rhind Papyrus says: Find a quantity such that when it is added to of itself the result is a 15. The mod
    12·2 answers
  • Simplify. Rewrite the expression in the form 5^n <br> 5^2*5^5
    8·2 answers
  • I need answers about those mathematic problem
    5·1 answer
  • Someone must have beaten me here. Devora thinks, dismayed by the empty treasure chest. Then, just as she's about to close the li
    10·1 answer
  • Kim has 12 ounces of water. She pours some waer into a bottle that holds exactly 1 cup. How many ounces of water are left over a
    13·1 answer
  • "Shaquille, Lyle, and Amanda collect stamps. Shaquille has one-half the number of stamps that Lyle has. Amanda has two-thirds as
    5·2 answers
  • Solve 5kx + 6 = 7kx for x.
    8·1 answer
  • 1)4(x– 12) + 19 = -89<br> 2)7+5(x-2)=12<br> 3)-4x-2(2x+10)=-116<br> 4)2(x+9)-4=68
    12·1 answer
  • Worksheet
    7·2 answers
  • Geometry question, I need help with this.
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!