Question

word problems on congruent triangles worksheet
1 for each of these given sets of triangles, state the rule that tells you that they are congruent. find the unknown values.
2 which of these pairs of triangles are similar? for the pairs that are similar, give the rule used to prove them similar.
3 solve for x in the given figures

Explanation:

Step1: Use the property of similar triangles (corresponding sides are proportional)

For \(\triangle ABD\sim\triangle CBD\), we have \(\frac{AD}{BD}=\frac{BD}{DC}\). Let \(AD = x\), \(BD = 12\), \(DC=16\).

Step2: Substitute values into the proportion

\(\frac{x}{12}=\frac{12}{16}\). Cross - multiply gives \(16x=12\times12\).

Step3: Solve for \(x\)

\(16x = 144\), then \(x=\frac{144}{16}=9\).

Step4: For \(\triangle AXZ\sim\triangle SYZ\), use the property of similar triangles (corresponding sides are proportional)

We know that \(\frac{AX}{SY}=\frac{XZ}{YZ}=\frac{AZ}{SZ}\). Here \(AX = 15 + y\), \(SY=y\), \(AZ=12 + x\), \(SZ=x\). Also, \(\frac{15 + y}{y}=\frac{12 + x}{x}\). But another way is to use the ratio of corresponding sides. Since \(\triangle AXZ\sim\triangle SYZ\), \(\frac{XZ}{YZ}=\frac{AZ}{SZ}\). We can also use the fact that \(\frac{15}{20}=\frac{12}{x}\) (by the property of similar - triangles and the ratio of non - overlapping sides)

Step5: Solve the proportion for \(x\) in the second part

Cross - multiply \(\frac{15}{20}=\frac{12}{x}\), we get \(15x=20\times12\). Then \(15x = 240\), and \(x = 16\)

Answer:

a) \(x = 9\)
b) \(x = 16\)