Answered

Given: $3 AB = 7 AD$ and $3 AC = 7 AE$

Prove: $\triangle ABC \sim \triangle ADE$



Complete the steps of the proof.



Statements		Reasons

1. $3 AB = 7 AD$		1. Given

2. $\frac{AB}{AD} = \frac{7}{3}$		2. Division property

3. $3 AC = 7 AE$		3. Given

4. $\frac{AC}{AE} = \frac{7}{3}$		4. [blank]

5. $\frac{AB}{AD} = \frac{AC}{AE}$		5. Substitution property

6. $\angle BAC$ and $\angle DAE$ are vertical angles		6. Definition of vertical angles

7. $\angle BAC \cong \angle DAE$		7. Vertical angles theorem

8. $\triangle ABC \sim \triangle ADE$		8. [blank]

Verified Answer :

  • Convert the given equations \(3 AB = 7 AD\) and \(3 AC = 7 AE\) into side-length ratios \(\frac{AB}{AD} = \frac{AC}{AE} = \frac{7}{3}\).
  • Use the Vertical Angles Theorem to establish that the included angles \(\angle BAC\) and \(\angle DAE\) are congruent.
  • Apply the SAS (Side-Angle-Side) Similarity Theorem to conclude that \(\triangle ABC \sim \triangle ADE\).
  • \(\boxed{\triangle ABC \sim \triangle ADE}\)

Explanation

  1. Proportionality of sides

The problem asks us to prove that \$\triangle ABC \sim \triangle ADE\$ given the equations \$3 AB = 7 AD\$ and \$3 AC = 7 AE\$.

First, we analyze the provided equations. We are given two relationships between the sides of the triangles:

  1. \$3 AB = 7 AD\$
  2. \$3 AC = 7 AE\$

These equations imply that the sides \$AB\$ and \$AD\$ are proportional, and \$AC\$ and \$AE\$ are proportional. Specifically, we can express these as ratios:

  • Dividing the first equation by \$3 AD\$, we get \$\frac{AB}{AD} = \frac{7}{3}\$.
  • Dividing the second equation by \$3 AE\$, we get \$\frac{AC}{AE} = \frac{7}{3}\$.

Since both ratios equal \$\frac{7}{3}\$, we can conclude that \$\frac{AB}{AD} = \frac{AC}{AE}\$ by the substitution property. This establishes that two pairs of corresponding sides are proportional.

  1. Congruence of included angles

Next, we consider the angles included between these sides. The angles are \$\angle BAC\$ and \$\angle DAE\$. According to the geometric configuration implied by the vertical angles, these two angles are opposite each other at the intersection point. By definition, \$\angle BAC\$ and \$\angle DAE\$ are vertical angles. The Vertical Angles Theorem states that vertical angles are congruent, therefore \$\angle BAC \cong \angle DAE\$.

  1. Applying the SAS similarity theorem

Finally, we apply the Side-Angle-Side (SAS) Similarity Theorem. This theorem states that if two sides of one triangle are proportional to the corresponding two sides of another triangle, and the included angles are congruent, then the two triangles are similar.

We have:

  1. \$\frac{AB}{AD} = \frac{AC}{AE}\$ (two sides proportional)
  2. \$\angle BAC \cong \angle DAE\$ (included angles congruent)

Therefore, by the SAS similarity theorem, we conclude that \$\triangle ABC \sim \triangle ADE\$. This completes the proof.

Examples

In architecture or surveying, you might need to determine the dimensions of a distant object or a field layout using a smaller, measurable model. If you know that two triangles share an angle and their side lengths around that angle are proportional, you can guarantee that the triangles are similar. This allows you to scale up your measurements from your model to the real-world structure, ensuring accurate proportions and design consistency.