<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

5.7: Angle-Angle-Side Triangles

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated4 minsto complete
%
Progress
Practice Angle-Angle-Side Triangles
Practice
Progress
Estimated4 minsto complete
%
Practice Now

You and a friend decide to go fly kites on a breezy Saturday afternoon. While sitting down to make your kites, you are working on make the best shape possible to catch the breeze. While your friend decides to go with a diamond shaped kite, you try out making a triangle shaped one. While trying to glue the kite together, you make the first and second piece lock together with a 70 angle. The angle between the first and third pieces is 40. Finally, you also have measured the length of the second piece and found that it is 22 inches long.

Your kite looks like this:

Is there a way to find out, using math, what the length of the third side will be?

AAS Triangles

The Law of Sines states: sinAa=sinBb. This is a ratio between the sine of an angle in a triangle and the length of the side opposite that angle to the sine of a different angle in that triangle and the length of the side opposing that second angle.

 

 

 

 

 

The Law of Sines allows us to find many quantities of interest in triangles by comparing sides and interior angles as a ratio. One case where we can to use the Law of Sines is when we know two of the angles in a triangle and a non-included side (AAS).

Solve using Law of Sines. 

Using GMN,G=42,N=73 and g=12. Find n.

Since we know two angles and one non-included side (g), we can find the other non-included side (n).

sin73nnsin42nn=sin4212=12sin73=12sin73sin4217.15

Continuing on from Example A, find M and m.

M is simply 1804273=65. To find side m, you can now use either the Law of Sines or Law of Cosines. Considering that the Law of Sines is a bit simpler and new, let’s use it. It does not matter which side and opposite angle you use in the ratio with M and m.

Option 1: G and g

sin65mmsin42mm=sin4212=12sin65=12sin65sin4216.25

Option 2: N and n

sin65mmsin73mm=sin7317.15=17.15sin65=17.15sin65sin7316.25

Solve using Law of Sines. 

A business group wants to build a golf course on a plot of land that was once a farm. The deed to the land is old and information about the land is incomplete. If AB is 5382 feet, BC is 3862 feet, AEB is 101,BDC is 74,EAB is 41 and DCB is 32, what are the lengths of the sides of each triangular piece of land? What is the total area of the land?

Before we can figure out the area of the land, we need to figure out the length of each side. In ABE, we know two angles and a non-included side. This is the AAS case. First, we will find the third angle in ABE by using the Triangle Sum Theorem. Then, we can use the Law of Sines to find both AE and EB.

ABEsin1015382AE(sin101)AEAE=180(41+101)=38=sin38AE=5382(sin38)=5382(sin38)sin101=3375.5 feetsin1015382=sin41EBEB(sin101)=5382(sin41)EB=5382(sin41)sin101EB3597.0 feet

Next, we need to find the missing side lengths in DCB. In this triangle, we again know two angles and a non-included side (AAS), which means we can use the Law of Sines. First, let’s find DBC=180(74+32)=74. Since both BDC and DBC measure 74, DCB is an isosceles triangle. This means that since BC is 3862 feet, DC is also 3862 feet. All we have left to find now is DB.

sin743862DB(sin74)DBDB=sin32DB=3862(sin32)=3862(sin32)sin742129.0 feet

Finally, we need to calculate the area of each triangle and then add the two areas together to get the total area. From the last section, we learned two area formulas, K=12 bcsinA and Heron’s Formula. In this case, since we have enough information to use either formula, we will use K=12 bcsinA since it is less computationally intense.

First, we will find the area of ABE.

ABE:

KK=12(3375.5)(5382)sin41=5,959,292.8 ft2

\begin{align*}\triangle DBC\end{align*}:

\begin{align*}K & = \frac{1}{2}(3862)(3862)\sin 32 \\ K & = 3, 951,884.6\ ft^2\end{align*}

The total area is \begin{align*}5,959,292.8 + 3,951,884.6 = 9,911,177.4\ ft^2\end{align*}.

Examples

Example 1

Earlier, you were asked to find the length of the third side of the triangle. 

Since you know two angles and one non-included side of the kite, you can find the other non-included side using the Law of Sines. Set up a ratio using the angles and side you know and the side you don't know.

\begin{align*}\frac{\sin 70^\circ}{x} & = \frac{\sin 40^\circ}{22} \\ x & = \frac{22 \sin 70^\circ}{\sin 40^\circ} \\ x & \approx 32.146\end{align*}

The length of the dowel rod on the unknown side will be approximately 32 inches.

Example 2

Find side "d" in the triangle below with the following information: \begin{align*}e = 214.9, D = 39.7^\circ, E = 41.3^\circ\end{align*}

\begin{align*}\frac{\sin 41.3^\circ}{214.9} = \frac{\sin 39.7^\circ}{d}, d = 208.0\end{align*}

Example 3

Find side "o" in the triangle below with the following information: \begin{align*}M = 31^\circ, O = 9^\circ, m = 15\end{align*}

\begin{align*}\frac{\sin 9^\circ}{o} = \frac{\sin 31^\circ}{15}, o = 4.6\end{align*}

Example 4

Find side "q" in the triangle below with the following information: \begin{align*}Q = 127^\circ, R = 21.8^\circ, r = 3.62\end{align*}

\begin{align*}\frac{\sin 127^\circ}{q} = \frac{\sin 21.8^\circ}{3.62}, q = 7.8\end{align*}

Review

In \begin{align*}\triangle ABC\end{align*}, \begin{align*}m\angle A=50^\circ\end{align*}, \begin{align*}m\angle B=34^\circ\end{align*}, and a=6.

  1. Find the length of b.
  2. Find the length of c.

In \begin{align*}\triangle KMS\end{align*}, \begin{align*}m\angle K=42^\circ\end{align*}, \begin{align*}m\angle M=26^\circ\end{align*}, and k=14.

  1. Find the length of m.
  2. Find the length of s.

In \begin{align*}\triangle DEF\end{align*}, \begin{align*}m\angle D=52^\circ\end{align*}, \begin{align*}m\angle E=78^\circ\end{align*}, and d=23.

  1. Find the length of e.
  2. Find the length of f.

In \begin{align*}\triangle PQR\end{align*}, \begin{align*}m\angle P=2^\circ\end{align*}, \begin{align*}m\angle Q=79^\circ\end{align*}, and p=20.

  1. Find the length of q.
  2. Find the length of r.

In \begin{align*}\triangle DOG\end{align*}, \begin{align*}m\angle D=50^\circ\end{align*}, \begin{align*}m\angle G=59^\circ\end{align*}, and o=12.

  1. Find the length of d.
  2. Find the length of g.

In \begin{align*}\triangle CAT\end{align*}, \begin{align*}m\angle C=82^\circ\end{align*}, \begin{align*}m\angle T=4^\circ\end{align*}, and a=8.

  1. Find the length of c.
  2. Find the length of t.

In \begin{align*}\triangle YOS\end{align*}, \begin{align*}m\angle Y=65^\circ\end{align*}, \begin{align*}m\angle O=72^\circ\end{align*}, and s=15.

  1. Find the length of o.
  2. Find the length of y.

In \begin{align*}\triangle HCO\end{align*}, \begin{align*}m\angle H=87^\circ\end{align*}, \begin{align*}m\angle C=14^\circ\end{align*}, and o=19.

  1. Find the length of h.
  2. Find the length of c.

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.7. 

Vocabulary

Angle Angle Side Triangle

Angle Angle Side Triangle

An 'angle angle side triangle' is a triangle where two of the angles and the non-included side are known quantities.

Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Subjects:
Grades:
Date Created:
Sep 26, 2012
Last Modified:
Jun 15, 2016
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the Modality. Click Customize to make your own copy.
Reviews
Help us create better content by rating and reviewing this modality.
Loading reviews...
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.TRG.532.L.1