Question

section 4.3 homework

1. whats wrong with this picture? explain.

review
in exercises 16 - 18, complete the statement of congruence.

1. $\triangle bar\cong\triangle$
2. $\triangle far\cong\triangle$
3. $\overline{hg}\cong\overline{he}\cong\overline{hj}$

$\overline{je}\cong\overline{ho}$
$\angle gho\cong\angle j$
$\triangle hej\cong\triangle$

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For the given triangle with sides \(a = 21\space cm\), \(b=25\space cm\), and \(c = 48\space cm\)
Check \(a + b>c\): \(21+25 = 46\space cm\)
Since \(46<48\), the sum of the two shorter sides (\(21\space cm\) and \(25\space cm\)) is not greater than the longest side (\(48\space cm\))

Step2: Analyze congruent triangles for 16

In the parallelogram - like figure (where \(EA\parallel BR\) and \(EB\parallel AR\)), for \(\triangle BAR\) and \(\triangle ABE\)
By SSS (Side - Side - Side) congruence (since \(EA = BR\), \(EB=AR\), and \(AB\) is common) \(\triangle BAR\cong\triangle ABE\)

Step3: Analyze congruent triangles for 17

In the given figure, for \(\triangle FAR\) and \(\triangle FNA\)
We have \(AR = FN\) (by given markings), \(\angle RAF=\angle NFA\) (by angle markings), and \(AF\) is common. By SAS (Side - Angle - Side) congruence \(\triangle FAR\cong\triangle FNA\)

Step4: Analyze congruent triangles for 18

Given \(\overline{HG}\cong\overline{HE}\cong\overline{HJ}\), \(\overline{JE}\cong\overline{HO}\), \(\angle GHO\cong\angle J\)
For \(\triangle HEJ\) and \(\triangle HGO\)
By SAS (since \(\overline{HE}\cong\overline{HG}\), \(\angle HEJ=\angle HGO\) (supplementary to equal angles in a way that can be derived from given congruences), \(\overline{EJ}\cong\overline{GO}\)) \(\triangle HEJ\cong\triangle HGO\)

Answer:

• For problem 8: The sum of the two shorter sides (\(21 + 25=46\)) is less than the longest side (\(48\)), so these side lengths cannot form a triangle.
• For problem 16: \(\triangle ABE\)
• For problem 17: \(\triangle FNA\)
• For problem 18: \(\triangle HGO\)